AOITaSV 


THE 

aval  atichitect's  and  shipbuilder's 
pocket-book 


M 


,.08  i&  flajYT-OSAWYA 


*-^niT    ,^^,=r^,jr^ 


A  D  VKTiTISKMK^^TS 


MARINE    PATTERN 

Duplex  Steam  Pumps 

To  Lloyd's  Requirements. 

VERTICAL    or    HORIZONTAL. 


HAYWARD-TYLER  &  CO.,  Ltd. 

99  Queen  Victoria  Street, 
LONDON,   E.G. 


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THE 

NAVAL  ARCHITECT'S  and  SHIPBUILDER'S 
POCKET-BOOK 

OF 

jfonnula^,    IRulee,   anb   ^ablee 

AND 

MARINE    ENGINEEirS    AND    SURVEYOR'S 
HANDY    BOOK    OF    REFERENCE 

BY  jjjf^j^ 

CLEMENT    MACKROW 

t  ( 

l^ATE   MEMBER  OF  THE  INSTITUTION  OF  NAVAL  ARCHITECTS 
T.ATR   LKCTIRER  ON  NAVAL  ARCHITECTURE  AT   THE  BOW  AND  BROMLEY   INSTITUTE 

AND 

LLOYD    WOOLLARD 

JtOYAL  CORPS  OF   NAVAL  CONSTRUCTORS 

MEMBER  OF  THE   INSTITUTION  OF   NAVAL   ARCHITECTS 

INSTRUCTOR   IN   NAVAL   ARCHITECTURE  AT  THE   R.N.  COLLEOE,   GREENWICH 

iElefacutlj  1SDiti0n 

TH0R0U(3i'HLY    heviscd 
WITH    A    SECTION    ON    AERONAUTICS 


NEW    YORK 

THE  NORMAN  W.  HENLEY  PUBLISHING  CO. 

132    NASSAU    STREET 

1916 


^ 


\ 


^v 


\ 


\ 


PRINTED   BY 

STEPHEN  AUSTIN  AND   SONS,   LTD. 

HERTFORD. 


PREFACE 

TO 

THE    ELEVENTH    EDITION. 


The  need  of  a  new  edition  of  this  Pocket-book  haf? 
arisen    through    the    continual    development    of    the 
science    of     Naval    Architecture,    and    the     tendency 
towards  standardization  and  regulation  of  parts  of  the 
structure  and  equipment  of  ships.    Very  many  changes 
have  been  introduced,  and  much  of  the  book  has  been 
)'e written,  but  where  possible  its  form  has  been  left 
unaltered.     Its  object  remains  the  same  as  that  stated 
in  the  Preface  to  the  original  edition,  viz.  to  condense 
into  a  compact  form  all  data  and  formulae  that  are 
o)-dinarily    required    by    the    Shipbuilder    or     Naval 
chitect. 
Amongst  the  new  matter  inserted,  it  is  believed  that 
the  section  on  Speed  and  Horse-power  will  be  useful  in 
enabling  ships  of  ordinary  form  to  be  approximately 
}>owered  from  the  data  therein  given  ;  a  brief  description 
of  modern  methods  of  powering  and  determining  forms 
suitable  from    a   propulsive    standpoint  has  also  been 
included.     The  necessity  for  economizing  weight  where 
possible  without  diminution  of  strength  has  led  to  the 
sections  on  Strength  of  Materials,  Kiveted  Joints,  and 
Stresses  in    Ships   being  considerably   extended.     In- 
formation concerning  British  Standard  Sections,  Screws, 
Iveys,  etc.,  has  also  been  added,  by  permission  of  the 
Engineering  Standards  Committee.     Finally,  two  new 
sections  on   Aeronautical   matters   will  be   of   service, 
not  onlv  to  those  engaged  in  that  modern  and  rapidly 

355437 


VI  PREFACE. 

developing  branch  of  engineering,  but  also  to  Naval 
Architects  on  account  of  the  kindred  nature  of  the 
subjects,  and  of  the  direfet  application  of  many  air  data 
to  questions  relating  to  the  resistance  of  bodies  in  water. 

The  remaining  subjects  treated,  which  were  also 
included  in  previous  editions,  have  now  been  brought 
completely  up-to-date ;  the  excision  of  obsolete  data 
has  enabled  the  new  matter  to  be  inserted  without 
increase  in  the  size  of  the  book.  The  new  tables  of 
logarithms,  etc.,  it  is  trusted,  "vvill  be  found  of  great 
practical  convenience  to  those  using  them. 

The  scope  and  extent  of  the  revision  were  arranged 
in  the  first  place  with  the  original  author ;  although, 
owing  to  his  death  before  the  comj)letion  of  the  work, 
the  absence  of  his  advice  and  experience  during  the 
later  stages  has  been  felt  and  regretted,  the  reviser  has 
had  the  benefit  of  securing  great  assistance  from  many 
sources  during  the  preparation  of  the  new  edition. 
Among  those  who  kindly  contributed,  the  reviser  is 
greatly  indebted  to  Mr.  A.  W.  Johns,  the  results  of 
w^hose  valued  experience  have  been  embodied  in  various 
parts  of  the  book ;  the  new  sections  *  Aerodynamics ' 
and  '  Aeronautics  '  are  entirely  due  to  him.  Considerable 
aid  in  the  treatment  of  Speed  and  Propellers  has  been 
rendered  by  Professor  T.  B.  Abell,  while  Mr.  E.  F. 
Atkinson  has  supplied  useful  data  concerning  small 
craft  and  tugs.  To  these,  and  to  many  others  to  whom 
reference  is  made  in  the  course  of  the  book,  the  reviser 
tenders  his  cordial  thanks.  He  also  trusts  that  the 
numerous  correspondents  who  have  offered  suggestions 
and  pointed  out  errors  in  previous  editions  may  be  led 
to  take  the  same  kindly  interest  in  the  present  revision. 

L.  W. 

Barnes  :    January  1,  1916. 


P  K  E  F  ax:  E 

TO 

THE    FIRST    EDITION. 


The  object  of  this  work  is  to  supply  the  great  want 
which  has  long  been  experienced  by  nearly  all  who  are 
connected  professionally  with  shipbuilding,  of  a  Pocket- 
Book  which  should  contain  all  the  ordinary  Formulae^ 
Rules,  and  Tables  required  when  working  out  necessary 
calculations,  which  up  to  the  present  time,  as  far  as  the 
Author  is  aware,  have  never  been  collected  and  put  into 
so  convenient  a  form,  but  have  remained  scattered 
through  a  number  of  large  w^orks,  entailing,  even  in 
referring  to  the  most  commonly  used  Formulae,  much 
waste  of  time  and  trouble.  An  effort  has  here  been 
made  to  gather  all  this  valuable  material,  and  to  con- 
dense it  into  as  compact  a  form  as  possible,  so  that  the 
Xaval  Architect  or  the  Shipbuilder  may  always  have 
ready  to  his  hand  rdiable  data  from  which  he  can  solve 
the  numerous  problems  which  daily  come  before  him. 
How  far  this  object  has  been  attained  may  best  be 
judged  by  those  who  have  felt  the  need  of  such  a  work. 
Several  elementary  subjects  have  been  treated  more 
fully  than  may  seem  consistent  with  the  character  of  the 
book.  This,  however,  has  been  done  for  the  benefit  of 
those  who  have  received  a  practical  rather  than  a  theo- 
retical training,  and  to  whom  such  a  book  ^s  this  would 
be  but  of  small  service  were  they  not  first  enabled  to 


viil  PREFACE. 

gather  a  few  elementary  principles,   by  which  means 
they  may  learn  to  use  and  understand  these  Formulae. 

In  justice  to  those  authors  whose  works  have  been 
consulted,  it  must  be  added  that  most  of  the  Eules  and 
Formulae  here  given  are  not  original,  although  perhaps 
appearing  in  a  new  shape  with  a  view  to  making  them 
simpler. 

There  are  many  into  whose  hands  this  work  will  fall 
who  are  well  able  to  criticise  it,  both  as  to  the  usefulness 
and  the  accuracy  of  the  matter  it  contains.  From  sucH' 
critics  the  Author  invites  any  corrections  or  fresh  mate- 
rial which  may  be  useful  for  future  editions. 


Sl^MMARY    OF    CONTENTS. 


-•<>• 


PAGES. 

Signs  and  Symbols      .... 

1-3 

Logarithms         ..... 

4-7 

Trigonometry     ..... 

7-13 

Curves  (Conic  Sections,  Catenary ,  Cycloid,  etc. 

)       13-18 

Differential  and  Integral  Calcnlns 

19-21 

Practical  Geometry     . 

.       22-36 

Mensuration  of  Areas  and  Perimeters  . 

.       36-49 

Mensuration  of  Solids 

.       49-59 

Centres  and  Moments  of  Figures . 

.       59-69 

Moments  of  Inertia  and  Eadii  of  Gyration 

.       69-75 

Mt'cbanical  Principles 

76-79 

('(■ntrp  of  Gravity         .... 

80 

Motioii         ...... 

81-83 

i  )ynamics  ...... 

.       84-88 

Hydrostatics        ..... 

88-89 

1  >i.s))lacement,  etc.        .... 

90-102 

Weight  and  Centre  of  Gravity  of  Shii)s 

102-109 

Ma])ilit\ 

110-143 

Waves         ...... 

143-149 

Rolling        ...... 

150-160 

Speed  and  Horse-power 

160-190 

Propellers  ...... 

190-197 

Speed  Trials  and  Tables 

197-207 

Sailing,  Force  of  Wind 

208-211 

-V                                        r>uiui»ii\.xvx    ur     v^UiN 

PAGES. 

Distances  down  Rivers 

.     212-221 

Weights  and  Dimensions  of  Materials  .         .     222-237 

Wire  and  Plate  Gauges 

.     238-240 

British  Standard  Sections    . 

.     241-254 

Notes  on  Materials 

.     255-259 

Weight  and  Strength  of  Materials 

.     260-263 

Admiralty  Tests,  etc.,  for  Materials      .         .     263-284 

Lloyds'  Tests  for  Materials . 

.     284-285 

Riveted  Joints  and  Rivets    . 

.     285-294 

Braced  Structures 

.     294-300 

Shearing    Forces    and     Bending 

Moments 

of  Beams     .... 

.     300-308 

Strength  of  Materials  and  Stresse 

s,  etc. : 

General       .... 

.     309-312 

Bending       .... 

.     312-327 

Compression 

.     328-331 

Shear           .... 

.     332-337 

Miscellaneous 

.      337-339, 344 

Keys  and  Wheel  Gearing 

.     340-344 

Longitudinal  Stresses  in  Ships 

.     345-352 

Mechanical  Powers 

.     353-363 

Notes  on  Steering,  Rudders,  etc. 

.     364-371 

Launching 

.     372-378 

Armour  and  Ordnance 

.     378-388 

Notes  on  Machinery    . 

.     388-390 

Notes  on  Design 

.     391-394 

Fans  and  Ventilation  of  Ships 

.     395-401 

Hydraulics 

.     402-404 

Heat            .         . 

.     405-406 

Aerodynamics  (Forces  on  Plates, 

etc.)  .         .     406-431 

SUMMARY   OF    CONTENTS.  XI 

Aeronautics  (Notes  on  Airships,  etc.)    .  431-448 

B.  of  T.  Regulations  for  Marine  Boilers,  etc.  448-466 

„            ,,     for  Motor  Passenger  Vessels  466-469 

„      for  Ships   ....  469-488 

Strength  of  Bulkheads         ....  483-487 
International    Regulations    for    Preventing 

Collisions  at  Sea           ....  487-490 

Tonnage     490-495 

B.  of  T.  Rules,  etc.,  for  Life-saving  Appliances  495-502 

„          ,,     for  Emigrant  Ships        .          .  502-505 

Lloyds'  Rules  for  determining  Size  of  Shafts  506 

„     for  Ships       ....  507-519 

„         ,,     for  Yachts  of  the  International 

Rating  Classes    .         .         .  520-526 

Anchors  and  Cables 527-533 

British  Standard  Pipes  and  Screws  533-537 

Ship  Fittings 538-556 

Seasoning  and  Measuring  Timber         .          .  557-561 
Miscellaneous  Data      .         .         .         .         .  562-566 
Dimensions  and  Weights  of  Blocks       .         .  567-571 
Weight   and  Strength  of  Hemp  and  Steel- 
wire  Rope 572-583 

Lloyds'  Rules  for  Yards,  Masts,  Rigging,  etc.  584-593 

Distances  of  Foreign  Ports  from  London       .  594 

Paints,  Caulking  Varnishes,  Galvanizing,  etc.  595-605 

English  and  Foreign  Weights  and  Measures  606-627 

Decimal  Equivalents  .         .         .         .         .  628-631 

Foreign  Money 632-633 

Discount  and  Equivalent  Price  Tables  .          .  634-635 

Useful  Numbers  and  Ready  Reckoners         .  636-638 


xu 


SUMMARY    OF    CONTENTS. 


Tables  of  Circular  Measure 

Tables  of  Areas   of  and  Circumferences   of 

Circles  ..... 

Tables  of  Areas  of  Segments  of  Circles 
Tables  of  Squares  and  Cubes  and  Roots  of 

Numbers      ..... 
Tables  of  Logarithms  and  Antilogarithms 
Tables     of     Exponential     and    Hyperbolic 

Functions    .         . 
Tables  of  Hyperbolic  Logarithms 
Tables  of  Natural  Sines,  Tangents,  etc. 
Tables  of  Logarithmic  Sines,  Tangents,  etc. 
Index        


639-64; 

642-65] 
652-654 

655-69^. 
700-70". 

708-71( 
711-715 
716-71S 
720-728 
725-742 


ii,||     MACKROW    AND    WOOLLAED'S 

POCKET   BOOK 


FORMULiE,   EULIS,   AND  TABLES 

FOR 

NAVAL  ARCHITECTS  AND  SHIP-BUILDERS. 

SIGNS  AND   SYMBOLS. 

'  The  following  are  some  of  the  signs  and  symbols  commonly 
used  in  algebraical  expressions: — 

=  This  is  the  sign  of  eqaality.  It  denotes  that  the  quantities 
so  connected  are  equal  to  one  another;  thus,  3  feet=  I  yard. 

+  This  is  the  sign  of  addition,  and  signifies  plus  or  more  ; 
thus,  4  +  3  =  7. 

—  This  is  the  sign  of  subtraction,  and  signifies  minus  or  less ; 
thus,  4-3  =  1. 

X  This  is  the  sign  of  multiplication,  and  signifies  multiplied 
by  or  into  ;  thus,  4x3  =  12. 

-^  or  /  This  is  the  sign  of  division,  and  signifies  divided  by  ; 
thus,  4-^2  =  2  or  4/2  =  2. 

0  {}  []  These  signs  are  called  brackets,  and  denote  that  the 
quantities  between  them  are  to  be  treated  as  one  quantity;  thus, 
5{3(4  +  2)-6(3-2)|=5(18-6)  =  60. 

This  sign  is  called  the  bar  or  vinculum,  and  is  sometimes 

used  instead  of  the  brackets  ;  thus,  3(4  +  2)  — 6(3  — 2)  x  5  =  60. 

Letters  are  often  used  to  shorten  or  simplify  a  formula. 
Thus,  supposing  we  wish  to  express  length  x  breadth  x  depth,  we 
might  put  the  initial  letters  only,  thus,  Ix  bx  d,  or,  as  is  usual 
when  algebraical  symbols  are  employed,  leave  out  the  sign  x 
between  the  factors  and  write  the  formula  l.b.d. 

When  it  is  wished  to  express  division  in  a  simple  form  the 
divisor  is  written  under  the  dividend ;  thus,  (x  +  y)-rz  =  ^^-iti^ 


Z  SIGNS   AND    SYMBOLS. 

!  , ,! ! ,  -'^  y  THese'-are  sigr^s  of  proportion;  the  sign  :  =  is 
to,  the  sign  ::  =  as  ;*  thus,"  1 ':  3^  '::  3  :  9,  1  is  to  3  as  3  is  to  9. 

<  This  sign  denotes  less  than  ;  thus  2  <  4  signifies  2  is  less 
than  4. 

>  This  sign  denotes  more  than ;  thus  4  >  2  signifies  4  is  more 
than  2. 

*.*  This  sign  signifies  because. 

.*.  This  sign  signifies  therefore.  Ux.:  ','  9  is  the  square  of 
3  .*,  3  is  the  root  of  9. 

~  This  sign  denotes  difference,  and  is  placed  between  two 
quantities  when  it  is  not  known  which  is  the  greater ;  thus 
(x  ~  y)  signifies  the  difference  between  .r  and  ij. 

,  ',  These  signs  are  used  to  express  certain  angles  in 
degrees,  minutes,  and  seconds  ;  thus  25  degrees  4  minutes  21 
seconds  would  be  expressed  25°  4'  21". 

JVbte. — The  two  latter  signs  are  often  used  to  express  feet  and 
inches;  thus  2  feet  6  inches  may  be  written  2'  6". 

\/  This  sign  is  called  the  radical  sign,  and  placed  before  a 
quantity  indicates  that  some  root  oi  it  is  to  be  taken,  and  a 
small  figure  placed  over  the  sign,  called  the  exponent  of  the  root, 
shows  what  root  is  to  be  extracted. 

Thus  ^/a  or  \^a  means  the  square  root  of  a. 
^a  „  cube  „ 

4/a  „  fourth      „ 

-^^    This  denotes  that  the  square  root  of  a  has  to  be  taken 

and  divided  by  h. 

-— -  This  denotes  that  h  has  to  be  divided  by  the  square 
Va 
root  of  a. 

'a+  ' 


y: 


This  denotes  that  the  square  root  oi  a+h  has  to  be 
a  +  d 

divided  by  the  square  root  of  a  +  d.     It  may  also  be  written 


ya  +  b 
a-\rd 


thus.     Z^^;-^.,  or 


h         \/a  +  h 


oc  This  is  another  sign  of  proportion.    Ex. :  acch;  that  is, 
a  varies  as  or  is  proportional  to  b. 

CO  This  sign  expresses  infinity ;  that  is,  it  denotes  a  quantity 
greater  than  any  finite  quantity. 

Q  This  sign  denotes  a  quantity  infinitely  small,  nought. 

L  This  sign  denotes  an  angle.    Ex. :    l_  abc  would  be  written, 
the  anftie  abc. 


SIGNS   AND   SYMBOLS.  -3 

L  This  sign  denotes  a  right  angle. 

_L  This  sign  denotes  a  perpendicular ;  as,  ab  Led,  i.e.  ah  is 
perpendicular  to  cd. 

A  This  sign  denotes  a  triangle;  thus,  Aabc,\.e.  the  triangle 
abc\ 

II  This  sign  denotes  parallel  to.  Ex.  :  ab  \\  cd  would  be 
written,  ab  is  parallel  to  cd. 

f  ox  F  These  express  a  function ;  as,  a  =/(.r)  ;  that  is,  a  is 
a  function  of  x  or  depends  on  x. 

/  This  is  the  sign  of  integration  ;  that  is,  it  indicates  that  the 
expression  before  which  it  is  placed  is  to  be  integrated.  When 
the  expression  has  to  be  integrated  twice  or  three  times  the  sign 
is  repeated  (ihus,JX,///)  ;  but  if  more  than  three  times  an  index 
is  placed  above  it  (thus,/"). 

D  ord  These  are  the  signs  of  differentiation  ;  an  index  placed 
above  the  sign  (thus,  d')  indicates  the  result  of  the  repetition 
of  the  process  denoted  by  that  sign. 

2  This  sign  (the  Greek  letter  sigma)  is  used  to  denote  that 
the  algebraical  sum  of  a  quantity  is  to  be  taken.  It  is  commonly 
used  to  indicate  the  sum  of  finite  differences,  just  as  the  symbol/ 
is  used  for  indefinitely  small  differences. 

g  This  sign  is  used  to  denote  the  acceleration  due  to  gravity 
at  any  given  latitude.  Its  value  is  about  32-2  in  foot-second  units 
and  981  in  C.G.S.  units. 

TT  The  Greek  letter  pi  is  invariably  used  to  denote  3-14159 ;  that 
is,  the  ratio  borne  by  the  diameter  of  a  circle  to  its  circumference. 

e  or  e  This  letter  is  generally  used  to  denote  2-71823,  which 
is  the  base  of  hyperbolic  or  Napierian  logarithms. 

I  n  or  n  !  termed  '  factorial  n  ',  where  n  is  a  positive  integer, 
denotes  the  product  of  the  series  n  (n-1)  (n-2)  .  .  .  2.1. 
Thus,  [£=3.2.  1  or  6;  and  [_5_=  5  .  4  .  3  .  2  .  1  =  120. 

}i  denotes  the  midship  section  or  midship  part  of  a  vessel. 

As  the  letters  of  the  Greek  alphabet  are  of  constant  recur- 
rence in  mathematical  formulas  it  has  been  deemed  advisable  to 
append  the  following  table  : — 


Ao 

Alpha. 

I    I 

Iota. 

Pp 

Rho. 

B)8 

Beta. 

K   K 

Kappa. 

2<r5 

Sigma. 

r7 

Gamma. 

A  \ 

Lambda. 

Tt 

Tau. 

A  5 

Delta. 

Mfi 

Mu. 

Tu 

Upsilon. 

E  e 

Epsilon. 

N  u 

Nu. 

*<^ 

Phi. 

z  C 

Zeta. 

E  1 

Xi. 

Xx 

Chi. 

Htj 

Eta. 

O  0 

Omicrou. 

^r^ 

Psi. 

@d 

Theta. 

nir 

Pi. 

Xl« 

Omega. 

LOGARITHMS. 


LOGARITHMS. 

Definition, — The  logarifchra  of  a  number  to  a  given  base  is 
the  index  of  the  power  to  which  the  base  must  be  raised  in 
order  to  become  equal  to  the  given  number.  Thus,  if  a*  =  N, 
X  is  called  the  logarithm  of  N  to  base  a. 

The  logarithms  naturally  occurring  in  analytical  formulas  are 
to  the  base  e,  which  is  equal  to  2-718  .  .  .  or  to  the  sum  of  the 

infinite  series  1  +  1  +  .-^  +  .-5-  +  ,—7-  +   .    .    .  ;     the  values 

L^      Li      L± 
of    the  logarithms    are    obtained   indirectly   from   the   formula 

3,2       ^3      ^i 
loge  {1  +  x)  =  X  -^  "^"q"~T"^   •  •  *      ^^^^  logarithms  are 

termed  Napierian  or  hyperbolic  logarithms  ;  their  values  are  given 
in  the  table  on  pp.  700-4. 

When  used  to  shorten  arithmetical  work,  *  common 
logarithms  '  are  employed,  having  10  as  their  base. 

Note. — The  logarithm  of  1  to  any  base  is  zero. 

To  Change  the  Base  of  a  Logarithm. 

Rule. — To  obtain  the  logarithm  of  a  number  to  base  7j 
from  that  to  base  a,  multiply  the  latter  logarithm  by  the 
logarithm  of  a  to  base  b,  or,  equally,  divide  it  by  the  logarithm 
of  b  to  base  a. 

The  logarithm  of  N  to  base  a  is  denoted  by  loga  N. 

.-.  logs  N  =  loga  N  X  log6  a  =  loga  N  -=-  loga  6. 

Since  loge  10  =  2-303  .  .  .  =    ,„^„ ,  the  hyperbolic  logarithm 

of  a  number  is  obtained  by  multiplying  its  common  logarithm  by 

Note. — The  integral  part  of  a  logarithm  is  termed  its 
characteristic,  and  the  decimal  part  its  mantissa. 

To   Find   the  Logarithm  of  a  Number. 

Rule. — The  characteristic  is  one  less  than  the  number  of 
digits  in  the  integral  part  of  the  number  ;  when  there  is  no 
integral  part,  the  characteristic  is  negative  and  is  numerically 
one  more  than  the  number  of  cyphers  between  the  decimal 
point  and  the  first  significant  figure.  In  the  latter  case  the 
minus  sign  is  placed  over,  instead  of  before,  the  characteristic. 

The  mantissa  is  invariably  positive  ;  its  value  for  numbers 
of  three  or  less  significant  figures  is  directly  obtained  from 
the  tables  on  pp.  700-4  ;    for  numbers  having  four  &ignificant 


LOGARITHMS.  O 

figures  the  tabular  differences  given  in  the  columns  on  the 
right  are  employed  thus — 
Ex.  1.— Find  log  of  42-G3.  Ex.  2.— Find  log  of  -7897. 

log        42-60  =  1-C294  log        -7890  =  1-8971 

tab.  diff.      3  =  3  tab.  diff.      7  = 4 

log        42-63  =  1-G297  log         -7897  =  1-8975 

Note.— The  tabular  difference  is  placed  under  the  extreme 
right-hand  figure  or  figures  of  the  mantissa. 

To  FIND  THE  ANTILOGARITHAf,  OR  THE  NUMBER  CORRESPONDING 

TO  A  GIVEN  Logarithm. 

EuLE. — From  the  tables  of  antilogarithms,  find  the  number 
corresponding  to  the  given  logarithm,  using  the  tabular 
differences  as  before  if  four  significant  figures  are  required. 
If  the  characteristic  is  positive,  the  decimal  point  is  so  placed 
that  the  number  of  digits  to  the  left  is  one  more  than  the 
characteristic  ;  if  negative,  the  number  of  ciphers  between 
the  decimal  point  and  the  first  significant  figure  is  one  less 
than  the  characteristic.     For  tables  v.  pp.  705-8. 

Ex.  1. — Find  the  number  Ex.  2. — ^Find  the  numbei 
whose  logarithm  is  5*8178.        whose  logarithm  is  3*1763 

antilog       8170  =  G561  antilog       1760  =  1500 

tab.  diff.  8  =      14  tab.  diff.  3  = 1 

antilog       8178  =  6575  antilog       1763  =  1501 

Number  required  is  657,500  to  Number  required  is   001501. 
four  significant  figures. 

To  Multiply  and  Di\n[DE  by  Logarithms. 

Rule. — Add  together  the  logarithms  of  the  numbers  in  the 
numerator,  and  those  of  the  numbers  in  the  denominator  ; 
subtract  the  latter  sum  from  the  former.  The  antilogarithm 
of  the  result  is  the  number  required. 

^     ,     ,    2       17-63         2-052 
Ex. :  Evaluate  -  x  -— -  x    ^,^^.,„^ 
3         35         '008175 

log    2         =    -3010  log  3            =    .4771 

log  17-63    =  1-2462  log  35          =  1-5441 

log     2-052  =    -3122  log  -008175  =  5-9125 

1-8594  1-9337 

subtract    i-9337 

antilog       1-9257  «        84-28— the  required  result  to 

four  significant  figures. 

Note. — It  is  advisable  to  perform  the  operations  of  addi- 
tion, multiplication,  etc.,  on  the  mantissa  and  characteristic 
separately. 


b  LOGARITHMS. 

Involution  and  Evolution  by  Logarithms. 

EuLE. — Multiply  the  log-arithm  of  the  number  by  the  index 
of  the  power  to  which  it  is  to  be  raised.  The  antilogarithni 
of  the  result  is  the  number  required. 

Ex.  1. — Find  the  cube  and  cube  root  of  'OSTS. 

log  -9873  =  i-9944  log  -9873  =  1-9944 

Multiply  by  3  =  -  3  +  2-9944 

i-9832  Divided  by  3) 

Antilog  i-9832  =  -9620  which  1-9981 

is  the  cube  of  -9873.  Antilog  1-9981  =  -9956,  which 

is  the  cube  root  of  -9873. 

Ex.  2.— Evaluate  (20-4)1  83. 

log  20-4  =  1-30C6,  say  1-310. 
To  multiply  this  by  1-83, 

log    1-310  =  -1173 
log    1-83    =  -2625 

•3798 

Antilog  -3798  =  2-397;  antilog  2-397  =  242-5,  the  re- 
quired result. 

Accuracy  of  Numerical  Calculations. 

In  general,  the  accuracy  of  the  result  of  a  numerical 
calculation  is  the  same  as  that  of  the  factor  liable  to  the 
greatest  proportional  error.  Exceptional  cases  arise,  viz., 
{a)  when  two  nearly  equal  numbers  are  subtracted  the  per- 
centage error  in  the  result  is  usually  greater  than  that  in 
either  of  the  numbers  ;  {b)  when  a  large  number  of  similar 
quantities,  such  as  the  ordinates  in  a  displacement  sheet,  are 
added,  the  individual  errors  of  measurement  tend  to  neutralize, 
and  the  accuracy  of  the  result  is  U3ually  greater  than  that 
of  its  component  factors  ;  (<?)  the  percentage  error  in  the 
n^^  power  of  a  number  is  n  times  that  of  a  number  ;  thus' 
in  the  cube  the  error  is  trebled,  but  in  the  cube  root  it  is 
divided  by  three.  Subject  to  these  qualifications  a  con- 
siderable saving  in  the  numerical  labour  of  a  calculation  may 
be  effected  by  limiting  the  number  of  significant  figures  at 
each  stage  to  that  appropriate  to  the  accuracy  of  the  result. 

In  calculations  affecting  the  weight,  buoyancy,  stability, 
speed,  strength,  etc.,  of  ships,  a  proportional  error  of  a^. 
least  0*1  per  cent,  i.e.  one  in  a  thousand,  may  generally  be 
expected  ;  three  or,  at  most,  four  significant  figures  are  suffi- 
cient in  such  cases,  any  additional  figures  being  meaningless 
and    redundant. 


TRIGONOMETRICAL    RATIOS.  7 

The  slide  rule,  which  mechanically  performs  the  operations 
of  multiplication,  division,  evolution,  etc.,  by  the  aid 
(virtually)  of  three-place  log^arithras,  is  usually  sufficiently 
accurate  for  the  majority  of  such  calculations  ;  tables  of 
logarithms,  trigonometrical  functions,  etc.,  to  four  (or  at 
most  five)  places  of  decimals  are  sufficient  to  perform  any 
calculations  in  which  rather  greater  accuracy  is  desired  and 
can  be  obtained. 


TRIGONOMETRY. 

The,  complement  of  an  angle  is  its  defect  from  a  right 
angle  ;  thus  if  A  denote  the  number  of  degrees  contained  in 
any  angle,  90°  —  a  is  the  number  of  degrees  contained  in  the 
complement  of  that  angle. 

The  supplement  of  an  angle  is  its  defect  from  two  right 
angles  ;  thus  180°  — a  is  the  number  of  degrees  contained  in 
the  supplement  of  that  angle. 


Fig.  1. 


Trigonometrical  Ratios. 

The  trigonometrical  ratios  of  an  angle  are 
defined  as  follows  :— Let  bag  (fig.  1)  be  any 
angle  ;  take  any  point  in  either  of  the  con- 
taining sides  and  from  it  draw  a  perpen- 
dicular to  the  other  side  ;  let  P  be  the  point 
in  the  side  AC,  and  pm  perpendicular  to 
AB  ;  let  A  denote  the  angle  BAG.     Then — 

perpendicular        PM 

sine  A  =^    ;,— — -— =  — 

hypotenuse  AP 


co-sme  A 


base 


AM 
AP 


hypotenuse 

perpendicular  _  PM 

base  AM 

base  _  AM 

perpendicular       PM 

hypotenuse     _  AP 

base  AM 

hypotenuse    _  AP 

perpendicular       PM 

versed  sine  A  =  1  -  cos  A 
co-versed  sine  A  =  1  -  sin  A. 

These  ratios  depend  only  on  the  angle,  and  are  independent 
of  the  position  of  the  point  P. 


tangent  A 


co-tangent  A  = 


secant  A  = 


co-secant  A 


8  MEASUREMENT  OF  ANGLES. 

Measurement  op  Angles. 

There  are  three  modes  of  measuring  angles,  viz. — 

1st.     The  sexagesimal  or  English  method. 

2nd.  The  centesimal  or  French  method. 

3rd.    The  circular  measure. 

The  sexagesimal  method  and  the  circular  measure  only  will 
be  dealt  with  here. 

The  Sexagesimal  Method. — In  this  method  a  right  angle  is 
supposed  to  be  divided  into  90  equal  parts,  each  of  which  parts 
is  termed  a  degree  ;  each  degree  is  divided  into  60  equal 
parts  called  minutes,  and  each  minute  is  divided  into  60  equal 
parts  called  seconds.    One  degree  16  minutes  15  seconds  or 

1°  16'  15",  is  therefore  equal  to  1  +  ^  +  -^wf^  or  1-271  degrees. 

The  Circular  Measure. — The  unit  of  circular  measure  is 
an  angle  which  is  subtended  at  the  centre  of  a  circle  by  an 
arc  equal  to  the  radius  of  that  circle.  It  is  called  a  radian. 
Such  an  angle  is  equal  to 

2  right  angles         180°         ^„  ^^         , 
.  =  30416  =  "-^   "^""-'y- 

The  circular  measure  of  an  angle  is  equal  to  a  fraction 
which  has  for  its  numerator  the  arc  subtended  by  that  angle 
at  the  centre  of  any  circle,  and  for  its  denominator  the  radius 
of  that  circle. 

Since  the  circumference  of  any  circle  is  27r  times  the  radius, 
four  right  angles  are  equal  to  2ir  radians.      Consequently  one 

right  angle  is  equal  to  —  radians. 

22         355 

Approximate  values  of  tt  are  3-1416  and  —  and  ry^ 

To  find  the  circular  7neasure  of  any  angle  expressed  in  degrees, 
minutes,  a7id  seconds. 

PtULE. — Multiply  the  measure  of  the  angle  in  degrees  by  ir, 
and  divide  by  180. 

Ex. :  Express  1°  16'  15"  or  1-271°  in  circular  measure. 

1   .071      V     TT 

-  ^jQfx —  =  -0222  circ.  meas. 

To  find  the  measure  of  any  angle  in  degrees,  minutes,  and 
seconds,  the  circular  measure  being  given. 

EuLE. — Multiply  the  circular  measure  of  the  angle  by  180. 
and  divide  by  ir. 


GENERAL   F011MLL.-S:.  s» 

Ex.  1.— Express  in  degrees,  etc.,  an  angle  the  circular  measure 

of  which  is -^  2^  X  180  ^  ^,Q, 

3  X  IT 
Tables    giving    the    circular    measure    of    angles    are    on 
pp.    639-41. 

General  Formula. 
sin^  0  +  cos^  e  =  1.  sec'^  0  =  1  +  tan'  9. 

cosec^  0  =  1  +  cot^  e. 
sin  (a  +  b)  =  sin  A  cos  B  +  cos  A  sin  B. 

cos  (a  +  b)  =  cos  A  cos  B  -  Sin  A  sin  B. 
sin  (a  -  b)  =  sin  A  cos  B  -  cos  A  sin  B. 

cos  (a  -  b)  =  cos  A  cos  B  +  sin  a  sin  B. 

„     .      A+  B  A  -  B 

sm  A  +  sm  B  =  2  sm  — ^ —  cos  — ^ — 

A  +  B  A  -  B 

cos  A  +  COS  B  =  2  COS  — ^ —  cos r — 


A  +  B     .      A 

sin  A  -  sm  B  =  2  cos  — x—  sm  — 


^     .      A  +  B     .     A 

COS  B  -  COS  A  =  2  sm — ^ —  sm 


2  2 

.  .         tan  A  +  tan  B 

*^°(^  +  ^^  =  l-tanAtanB 

,  .         tan  A  -  tan  B 

*^"  (^  -  ^)  =  1  +  tanAtanB 

sin' 2a  =  2  sin  a  cos  a.  sin  3a  =  3  sin  a  -  4  sin'  A. 

COS  2a  =  cos^  A  -  sin^  A.       cos  3A  =  4  cos^  A  -  3  cos  A. 

.A  .     ^   /I  -  cos  A  A  .     ^  /I  +  COS  A 

sm- 


.     A  /I  -  COS  A  A  ,     A  /I  + 

+  V  — o —      ^°^^  =  +  V  — 


2—^2  2—^2 

A  2i  \  —  t^  2i 

If  t  =  tan  -,  sin  A  =  -^  ,  ^2  \  cos  A  =  j^qjp  ;  tan  A  =  Y~t^ 

And  when  A,  B,  c  are  the  three  angles  of  a  triangle, 
A  +  B  +  c  =  IT  radians  or  two  right  angles  ; 
and  sin  (a  +  b)  =  sin  (tt  -  c)  =  sin  c. 

When  A  is  any  angle, 
sin  (  -  a)  =  -  sin  A.  cos  ( -  a)  =  cos  A. 

tan  ( -  a)  =  -  tan  A. 
sin  (90°  -  a)  =  cos  A.  cos  (90°  -  a)  =  sin  A. 

tan  (90°  -  a)  =  cot  A. 
sin  (90°  +  a)  =  cos  a.  cos  (90°  +  a)  =  -  sin  a. 

tan  (90°  +  a)  =  -  cot  A. 
sin  (180°  -  a)  =  sin  A.  cos  (180°  -  a)  =  -  cos  A. 

tan  (180°  -  a)  =  -  tan  A. 
sin  (180°  +  a)  =  -  sin  A.        cos  (180°  +  a)  =  -  cos  A. 

tan  (180°  +  a)  =  tan  A. 


JO 


FUNCTIONS,   PROPERTIES  OF  TRIANGLES. 


The  algebraic  formulss  for  the  sine  and  cosine  are- 


—  4- —  _ 
3!  "^5! 


eA  V 


e-A\/-i 


cos  A 


V-i  +  e--^V-i 


1    -    —  +  — 


where  A  is  in  circular  measure. 


2' 


Where   A   is   small,    sin  A  =  tan   A  =  A  ;    cos   A  = 
A^ 
sec  A  =  1  +  — 

Tables  of  the  trigonometrical  functions  are  given  on  pp.  716-19. 

Inverse  Functions. 

If  sin  a  =  X,  then  a  =  sin~^a;. 
If  cos  a  =  y,  then  a  =  cos~^y. 
And  so  on. 

Note. — sin"~^x  is  read  '  inverse  sine  x\  etc. 

Logarithmic  Functions. 

The  logarithms  of  the  sines,  cosines,  etc.,  are  denoted  log 
sin,  log  cos,  etc.,  and  their  values  are  given  on  pp.  720-3. 
For  convenience  the  characteristic  is  in  each  case  increased  by 
the   number    10. 


Properties  of  Triangles. 
Fig.  3. 


Fig.  4. 


Note. — The  sides  opposite  the  angles  A,  B,  C  respectively  will 
be  denoted  by  the  letters  a,  b,  c.  The  angle  BDA  in  figs.  2  and  3 
is  a  right  angle. 

In  fig.  2,  where  B  and  c  are  acute  angles,  we  have — 


AD        AD 

AD         AD 

Sin  c  =  —  =  -,- 
AC         b 

sin  B        AD    ,    AD 

sin  G        c     '     b   ~ 

6 

c 

TRIANGLES.  1 1 

In  fig.  3,  where  C  is  an  obtuse  angle,  and  in  fig.  4,  where  c  is 

a  right  angle,  the  proof  is  similar. 

.     ,  , ,        r       .  ,  .       ,    sin  A     sin  B     sin  c 

And  therefore  in  any  triangle =  —~ —  = . 

a  o  c 

Also  cos  A 


2.bo 


•.in  ^-  A  A*-*X*-^)  •      cos  ^=  A  /<'-''\ 
sin^-y ^^ ,      cos  2     V"*6' 

A  /(s-h)(s-c)         .  2    ,- — — - 

where  2«  =  <z  +  J  +  <?. 

,  ,       B— c     h—c      .  A 

a  =  b  cos  c  +  c  cos  B  ;  tan  — -—  =  , cot  -. 

2       b  +  o         2 

Area  of  triangle  =  -^  sin  A  =  \'s{s-  a)  (s  -h)  (s  -  c) 

Solution  of  Triangles. 

Every  triangle  has  six  elements — three  sides  and  three 
angles.  If  any  three  of  these  be  given  (provided  they  be  not 
the  three  angles)  the  triangle  can  be  completely  determined. 


Right-angled  Triangles. 

Let  C  be  the  right  angle,  and  therefore  c  the  hypotenuse, 
(i.)  Given  hypotenuse  (c)  and  one  side  (a). 

b  =  s/'i^^a\  tan  B  =  K  and  A  =  90°  -  B. 


/a?-  +  ¥,  tan  B  =    ,  and  A  =  90°  -  b. 


(ii.)  Given  the  two  sides  (a  and  J). 
b 
a 

(iii.)  Given  an  angle  (b)  and  one  of  the  sides  {a). 

b  =  a  tan  B,  c-a  sec  B. 
(iv.)  Given  an  angle  (b)  and  the  hypotenuse  (c). 
a  =  ('  cos  B,  b  =  e  sin  B,  A  =  90°  —  B. 


Any  Triangles. 

(i.)  Given  the  three  sides,  a,  b,  and  c. 

tan- 


A_      /(s-bXs-a')  B_      /(^-^)(<-.0 


C=180°-A-B, 

where  2s  =  a  +  b  +  c. 


12 


MEASUREMENT   OF   HEIGHTS    AND   DISTANCES. 


(ii.)  Given  two  sides,  h  and  c,  and  the  included  angle  A. 


,       B— c     h—o      ,A 
tan =  ; cot  -. 

2        b  ^G  2 


B  +  C 

"2^ 


90^- 


T^  B-CjB  +  C  ,  j^  -,  7.    Sin  A 

From       and— ^—  we  can  get  B  and  C  ;  and  a  =  h    - —  . 

2  2  sin  B 

(iii.)  Given  two  sides,  h  and  c,  and  the  angle  B  opposite  to 
one  of  them. 

sin  c  =  -  sin  B.     We  thus  obtain  c  ;  and  A  =  (180  -  B  -  C). 
h 


Also 


-,  sm  A 

a  =  b    . — . 

sm  B 


As  there  are  generally  two  angles  between  0°  and   180° 

whose  sine  is  -  sin  B,  two  values  of  c  are  often  admissible,  and 

b 
sometimes  two  triangles  can  be  constructed. 

(iv.)  Given  one  side  and  two  angles,  a,  b,  and  C. 

-,  orxo  X.        sin  B  sin  0 

A  =  180°  -  B  -  c  ;  b  =  a \  c  =  a 

sm  A  sm  A 

(v.)  When  the  three  angles   only  are  given,  the  absolute 

magnitude  of  the  sides  cannot  be  determined,  but  their  ratios 

.         a  b  G 

are  given  by  -; =  -; —  =  -; 

sm  A     sin  B     sin  c 


Table   giving   the   Signs  and   Values  of  the 
Trigonometrical  Ratios  for  Certain  Angles. 

Ratios 

0° 

Signs    30°  1 

Signs  j  45° 

Signs 

60° 

Signs 

90° 

I 
0 

OD 

0 

00 

1 

Signs 

120° 

2 
1 
2 

V3 

1 

v/3 
2 

2 

Sine 
Co-sine 
Tangent 
Co-tangent 
Secant 
Co-secant 

0 
1 
0 

CO 

1 

00 

+ 

+ 
+ 
+ 
+ 

1 
2 

V3 
2 

1 

^n 

V3 

2 

V3 
2 

+ 

-1- 

-t- 
+ 
+ 
+ 

1 

V2 

1 

1 

V2 
V2 

+ 
+ 
+ 
-t- 
+ 
■¥ 

2 
I 
2 

V3 

1 
V3 

2 
2 
V3 

+ 
+ 

+ 
4- 

-h 
+ 

Ratios 

Signs 

135° 

Signs 

150° 

Signs 

180° 

Signs|270° 

Signs 

360° 
0 
1 
0 

00 

1 

00 

Sine 

Co-sine 

Tangent 
Co-tangent 
Secant 
Co-seoant 

+ 

1 

V2 

1 

1 
1 

!    V2 

+ 
-f 

1 

2 
,_/_3 

2 

1 
V3 

2 

+ 

4- 
+ 

4- 

0 

1 
0 

00 

1 

X 

- 
+ 

1 

0 

00 

0 

00 

1 

+ 

HYPERBOLIC  FUN'CTIONS,  PARABOLA. 


13 


Hyperbolic  Functions. 
The  hyperbolic  functions  are  used  in  connection  -with  the 
catenary  ;    they  are  six  in  number,  and  are  represented  by 
affixing   h  to   the   symbols   of   the   trigonometrical   functions. 
They  are  determined  by  the  following  formulae  : — 


Binh 

..-- 

cosh 

tanha;  = 

sinh  X 
coshtc 

Rfifib  T  = 

1 

=  -£- 


14  ^ 


coth  X  = 


cosech  X 


cosh  X 

sinh  X 

1 


1_ 

tanh  X 


cosh  a;  sinhar. 

'Note. — All  formulae  connecting  sin,  cos,  and  tan  can  be 
converted  into  the  corresponding  formulse  for  sinh,  cosh,  tanh  by 
changing  sin  a;  to  V  - 1  sinh  a;,  cos  x  to  cosh  x,  and  tan  x  to 
V  - 1  tanh  X  ;  thus  cosh^  x  -  sinh^  a;  =  1  ;  sech'-^  x  +  tanh"  x 
=  1  ;  etc. 

The  values  of  sinh  x,  cosh  x,  e*,  and  e~*  are  given  in  the 
tables  on  pp.  708-10. 

CURVES. 

CONIC   SECTIONS. 

Definition.— The  locus  of  a  point  which  moves  so  that  its 
distance  from  a  fixed  point  is  always  in  a  constant  ratio  to  its 
perpendicular  distance  from  a  fixed  straight  line  is  called  a  conic 
section. 

The  fixed  point  is  called  the  focus,  the  constant  ratio  the 
eccentricity,  and  the  fixed  straight  line  the  directrix. 

The  straight  line  passing  through  the  focus  and  perpendicular 
to  the  directrix  is  called  the  axis. 

Parabola. 

The  conic  section  is  called  a  parabola 
when  the  eccentricity  is  equal  to  unity. 

In  fig.  5,  F  is  the  focus,  AB  the 
directrix,  AX  the  axis,  o  the  intersection 
of  the  curve  with  the  axis,  OY  a  line 
perpendicular  to  AX,  and  P  any  point  on 
the  curve  ;  then  PQ  =  PF. 

The  equation  of  the  curve  with  OY 
and  ox  as  axes  is 

2/^  =  4  ax,  where  AO  =  OF  =  a. 

A  parabola  may  also  be  defined  as  the  section  of  a  cone  cut 
by  a  plane  parallel  to  one  of  the  slant  sides. 


Fig.  5. 

A  oUf                               X 

14 


ELLIPSE,  HYPERBOLA. 


Ellipse. 

The  conic  section  is  called  an  ellipse 
when  the  eccentricity  is  less  than  unity. 

In  fig.  6  CD  is  the  directrix,  F  the 
focus,  aa'  the  major  axis,  o  the  middle 

point  of  aa',  and  bb'  the  minor  axis,   Qj "7 

down  through  o,   perpendicular  to   the    J /_ 

axis,  and  P  any  point  on  the  curve  so    Dl       A*. 

PF 
that  ~=  the  eccentricity  e.     The  equa- 
tion to  the  curve  with  oa',  oy  as  axes  is — 

a" 
Also  OA  =  oa'  =  a  ;  OB  ^  ob'  =  6 ;  a^ 

a 
ae;  od  =  - 


1. 


Fig.  6. 


a'  e' 


OF 


An  ellipse  may  also  be  defined  as  the  intersection  of  a  cone 
by  a  plane  passing  through  its  slant  sides,  but  not  perpendicular 
to  the  axis. 

Hypekbola. 
The  conio  section  is  called  a  hyperbola  when  the  eccentricity 
is  greater  than  unity. 

In  fig.   7  ab  is  the   directrix,   F   the   focus,  xx'  the  axis, 
cc'  the  points  where  the  curve  intersects  the  axis,  OY  a  line 
Fig.  7. 
Y, 
i   B 


O  A 


drawn  through  the  middle  point  of  CC'  perpendicular  to  the  axis, 
and  P  any  point  on  either  branch  of  the  curve. 

PF 

Then  —  =  the  eccentricity  e. 
PQ 


Taking  ox  and  OY  as  axes  the  equation  to  the  curve  is— 

=  1. 

a 


£_  _  V- 
Also  oc  =  oo'  =  a  :  a-  +  y^  ■■ 


a' 


OF 


ae:  oa  = 


CATENARY. 


16 


If  the  sides  of  a  cone  be  produced  beyond  the  vertex  so  as 
to  form  a  second  cone  with  the  same  axis  as  the  first,  and  these 
two  cones  be  cut  by  a  plane,  the  section  will  be  a  hyperbola. 

If  b  be  made  equal  to  a  in  the  above  equation,  it  becomes 
x"'  -'^  =  a-,   which  is  a  rectangular  hyperbola.      By  turning 

Fig.  8. 
Y 


the  axis  through  an  angle  of  45°,  the  equation  becomes  of  the 

form  xy  =  c^  {fig.  8),  where  c^  =  \a'^ 

Catenary. 
(See  pp.  27  and  28  for  method  of  construction.) 
If  a  uniform  chain  be  freely  suspended  from  two  points,  A 
and  B,  the  curve  in  which  it  will  hang  is  termed  a  common 
catenary  ;  the  parameter  DC  is  equal  to  the  length  of  a  piece  of 
the  chain  whose  weight  is  equal  to  the  tension  at  the  lowest 
point  C  in  the  curve. 

FlQ.  s). 


The  directrix  OX  is  a  horizontal  line  drawn  through  the 
extremity  0  of  the  parameter. 

The  tension  at  any  point  p  in  the  carve  is  equal  to  the 
weight  of  a  piece  of  the  chain  whose  length  is  equal  to  the 
ordinate  PM. 

Equaiimis  to  the  Catenary  (see  fig.  9). 
Take  ox  (horizontal)  and  CD  (vertical  through  C  the  lowest 
point)  as  axes. 


16  EQUATIONS    TO    THE    CATENARY. 

X  =  abscissa  CM.  y  —  ordinate  pm.  c  =  parameter  00. 
s  =  length  CP  of  chain,  w  —  weight  of  chain  per  hnear  unit  run. 
T  =  tension  at  P.  0  =  angle  to  horizontal  of  chain  at  P. 
e  =  base  of  hyperbolic  logarithms  =  2-718  .  .  . 

y  =  c  cosh^  =  |(e^  +  e "0  =  ^  ^^^  +  «')' 

s  =  c  sinh|  =  ^(e*"^  -  e~^^  =  V  (?/^  -  c^). 

1  =  11)0  cosh—  =  wy.       Dip  (dc  in  fig.)  =  c  (cosh—  _  i), 
c  c 

tan  <t>  =  sinh  —  =  —         sec  *  =  cosh  —  =  — 
c      c  c      c 

The  values  of  the  hyperbolic  functions  (sinli  x,  cosh  a:,  etc.) 
are  tabulated  on  pp.  708-10.  Examples  showing  their 
application  to  the  catenary  are  given  below. 

Approximate  Equations  for  flat  piece  of  chain,  nearly 
horizontal. 


5  =  dip  DC  =  7/-c;   a;  =  i  span. 

'""2     c    '24c''~'»'V2 

\    x^      1  .      s*  -  82 
^  =  2     2 

1    a:^  .    2    82 


2-c  +2i?=v(2^(^-^); 

,   1  ,      s*-82  a/_A_ 

+  6 '="^^=^^67^::^ 


^  =  ^  +  6    ^^=^+  3    X 
Tension*  =  ^  total  weight  x  span  -f  sag  at  centre. 

-^oie.— When  the  points  of  support  are  in  the  same 
horizontal  plane,  the  catenary  is  symmetrical  about  a  vertical 
line  ^passing  midway  between  them,  and  the  preceding  formulae 
can  be  directly  employed  to  determine  the  particulars  of 
the  curve. 

Ex.  1. — A  chain  weighing  15  lb.  per  foot  run  is  suspended 
between  two  points  at  the  same  level  and  100  feet  apart.  The 
dip  is  observed  to  be  40  feet.  Determine  the  length  of  chain, 
the  maximum  tension,  and  the  inclination  at  the  supports. 

Dip  =  c  (cosh—  -  l).     Here  dip  =  40 ;  ^=—^  =  50. 

Hence  40  =  c  (cosh—  -  l).  By  trial,  from  the  tables  (p.  709), 

c  =  36  approximately. 

Length  of  chain  =  2s  =  2c  sinh—  =  135  feet  approximately. 

c 

*  On  substituting  '  pressure '  for  '  weight ',  this  is  applicable  to  a  rope 
or  net  under  uniform  pressure  when  sag  is  moderate. 


EQUATIONS    TO    THE    CATENARY. 


17 


Maximum  tension  occurs  at  supports  and  is  given  by — 

X 


T  =  wc  cosh 


11501b. 


Angle  at  supports  =  <p 


sec~*  cosh  —  =  62®. 
c 


Ex.  2. — The  chain  in  the  preceding  example  is  tightened 
u-itil  the  length  suspended  is  reduced  to  120  feet.  To  determine 
the  dip — 


47 


Ex.  3. — If  the  chain  in  example  2  is  tightened  further 
until  the  dip  is  reduced  to  9  feet,  determine  the  length,  and 
the  t<insion. 

Using  the  approximate  formula;  for  a  flat  chain — 


s  =  csinh-. 
c 

Here  s  = 

60 ;  a;  =  50. 

Hence  60  =  c 

•  u  50 
smh  — . 
c 

By  trial,    from  the 

table?, 

c 

approximately. 

Dip  =  c  (cosh 

7-0  = 

29-2  ft. 

Length  =  2s 
Here  a;  =  50,  5  =  S 
Tension  =^  wc  =  w 


2^  +  5- 
3  X 


.     Hence  length  =  102  feet  approximately. 


Formulce  for  the  Catenary  beUceen  tivo  points  not  in  the  same 
horizontal  plane. 

Take  axes  as  before  (fig.  10), 

let  A,  B  be  the  points  of  support 

8  =  total  length  of  chain  ACB. 

5  =  vertical  distance  am  between 

A  and  B. 
a  =  horizontal     distance     MB    be 

tween  A  and  B. 

V  =  height    of    A    above    axis    ox 
2« 


Vs 


6-  =  2c  sinh 


2c 


2  c  sinh  —  cosh  — - — 
2c  2c 


2  c  sinh  r-  sinh —  ^ 
2  c  2  c 


2u 


c  cosh 


u 


Ex.  1. — A  chain  of  length  100  feet  is  suspended  between 
supports  distant  50  feet  horizontally  and  69  feet  vertically. 
Determine  the  position  of  its  lowest  point,  and  the  maximum 
tension  (wciglit  10  lb.  per  foot  run). 


18 


CYCLOIDAL   CURVES,    EVOLDTES    AND    INVOLUTES. 


Here  S  =  100 ;  a  =  50  ;  6  =  60. 
By  trial  from  the  above  formulse,  using  the  tables. 
c  =  14-2;  u  =  15-2;  v  =  23. 


Maximum  tension  (at  b)  =  w  c  sinh 


=  820  lb. 


Note. — If  a  is  negative,  the  lowest  point  of  the  catenary 
occurs  outside  the  point  of  support  A  ;  in  that  case  no  part 
of  the  chain  is  horizontal. 

Cycloidal  Curves. 
Definition. — If  a  circle  be  made  to  roll  without  slipping 
on  a  straight  line,  the  locus  of  a  point  P  (fig.  11)  on  its  cii-cum- 
ference  is  the  cycloid  mnm',  and  that  of  any  point  Q  inside 
the  circle  is  the  trochoid  rsr'. 

The  cycloid  meets  the  straight  line  ab  at  a  series  of  cusps 
mm'  .  .  .  ,  corresponding  to  the  positions  when  the  point  P  is 
vertically  above  the  centre  0  of  the  rolling  circle  ;    on  the 
trochoid  these  become  '  crests  *  similar  to  those  in  the  section 
of    a    wave-surface,    which    this    curve    is    found    closely    to 
resemble  ;   intermediate   between  M  and  m'  is  the  *  trough '  s 
which  has  a  smaller  curvature  than  the  '  crest  \     With  co- 
ordinates ox,  oy,  as  shown,  the  equations  to  the  cycloid  is— 
X  =  B.  {0  -  sin  6),  y  =  -R  cos  6,  and  to  the  trochoid  is — 
X  =  B.  0  -  r  sin  6,  y  =  r  cos  0 ; 
where  0  is  the  angle  pcl,  r  =  PC,  and  r  =  QC. 
Fig.  11. 


A           1 

m!' 

M'  e 

(^ 

^    ^V 

.-'Ax 

V. 

[J.-o^ 

-._s..... 

^.  - 

The  curve  described  by  a  point  on  the  circumference  of 
a  circle  rolling  on  the  exterior  of  another  circle  is  termed  an 
epicycloid  ;  when  rolling  on  the  interior  of  the  second  circle 
it  is  termed  a  hypocycloid. 

E VOLUTES  AND  INVOLUTES. 

Definition. — If  a  curve  be  drawn  passing  tlirough  the 
centres  of  curvature  at  various  points  at  a  curve,  the  new  curve 
is  said  to  be  the  evolute  of  the  original  curve  ;  conversely 
the  original  curve  is  termed  the  involute  of  the  derived  curve. 

The  involute  of  a  curve  is  also  derived  by  wrapping 
a  thread  around  the  circumference  of  the  curve  ;  the  path 
described  by  a  j)oInt  on  the  thread  as  it  is  wound  or  unwound 
is  the  involute. 


DIFFERENTIAL  CALCULUS. 


19 


DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

DlFFEllENTIAL  CaLCULUS. 

Definition. — A  quantity  is  said  to  be  a  function  of  anoilicr 
when  its  value  depends  upon  the  value  of  the  other. 

Thus,  x^,  sin  x,  c*  are  functions  of  x  ;  xij  is  a  function 
X  and  of  y  ;  and  so  on.  A  function  of  x  is  denoted  by  f{x). 
The  differential  coefficient  of  a  function  (y)  with  respect  to 
a  variable  (a;)  is  the  rate  of  increase  of  y  corresponding  to  an 

indefinitely  small  increment  of  x.     It  is  denoted  by  -r-  or  j'{x). 

Thus,  the  speed  of  a  ship  is  the  differential  coefficient  of 
the  distance  travelled  with  respect  to  the  time  occupied. 

Values  of  the  differential  coefficients  with  respect  to  x  of 
some  functions  of  x  are  given  in  the  table  below  :  — 


y  or  fix) 


uv 

Binx 

COSJC 

tanx 
cotx 

sec  a; 
cosec  X 
f(u) 

sin~'x  or-co3~'a; 

tan~^x  or— cot"'x 

seo"'x  or— cosec-'x 

vers'^x  or— covers  "'x 


n  X"   * 

dv  ,     du 
"rix  +  ^'d^ 

cosx 
-sin  as 

sec^aj 
-cosec^ac 

sec  X . tan  x 

- cosec X.  cot  X 
du  ^  d  fin) 
dx      du 

1 

Vl-x2 

_1 

1  +  X« 
1 


y  or  fix) 


J^orf'ix) 


u 

V 

sinhas 
cosh  X 
tanh  X 
cothx 
Bechx 
cosech  X 

log^X 
sinh"ix  or 

log(x+ Vx2  +  1) 

cosh  "  ^  X  or 

log(x+ Vx2-l) 

tanh" ^x  or 
coth"^xor 


Uog 


-I 


a^  loggO 

du       dv 

'^dx-"'di 

coshx 

sinhx 

sech2  X 
•cosech^x 

sechx  .  tanhx 

■cosech  X  .cothx 

1 


1 

1-X2 


_J 

l-x-i 


hote. —  The  differential  coefficient  of  -f-  with  respect  to  x  is 
d^v  dx  ^ 

denoted  ^,  and  is  termed  the  second  differential  coefficient  of  y. 

ArrLicATiON  to  Curves. 
The  equation  of  the  tangent  at  a  point  (x,  y)  on  a  curve  is 

dv 
That  of  the  normal  to  the  cur^-e  is  X  -  a;  +  (y  -  j/)  ^  =  0, 


20         Ari'IJCATION  TO  CURVES,  INTEGRAL  CALCULUS. 


The  angle  <p  made  by  the  tangent  with  the  x  axis,  and  the 
perimeter  s  measured  from  any  fixed  point  on  a  curve  are 
connected  by — 

dy      .    ^      dy  dx     /dx\^    /dy\i 


ax     /axY    /ay\2    , 
nt  is  given  by — 

-l>^  X— fo- T .79.. —    "1  mZ 


The  radius  of  curvature  {p)  at  a  point  is  given  by — 


d^/ 
dx^ 


dx^ 


—  d^ 

dy" 

When  tlie  curve  is  tangential  to  the  x  axis  at  the  origin,  then 

x^ 
P  is  equal  to  —  when  x  and  y  are  very  small. 

Integral  Calculus. 

Definition. — If  ^  is  the  differential  coefficient  of  a  funcfcioD 
2  Vr-ith  respect  to  x,  then,  conversely,  z  is  termed  the  integral 
o?  y  with  respect  to  cc,  being  denoted  Jy  dx. 

A  tabl6  of  integrals  frequently  required  is  appended. 


dx 


«  or  \  !/ . 


dx 


dz 


z  ov  \  y  .  dx 


a* 

cos  X 
cot  X 

1 


1 

x'i 

1 

-a^ 

Va;2  + 

a2 

Va2- 

a;2 

Va;2-a2 
sec  X 

1 

a2-a;2 


tt^/logr^  a 

sin  a; 
log  sin  X 


.    _i  x 

sm   1  - 

a 


a 

sinh  "  ^ 


a; 

einx 
tan  a; 
_J 

1 


;Va2: 


logtan(^+|) 

1   ,      g+a; 
2^  ^°^  ^a: 


V2aa:-a;2 

1 
Va;2-a2 

Va2  +  x2 

cosec  a; 
1 


log  a; 


—cos  X 
log  sec  X 


a  a 


cosh"' — ■ 
a 


xVrt''i  +  a;2  .  a2 


2^^"^'' a 


log  tan  ^ 
1   ,      x-a 


n\v.dx-\{f^.]v.dx)dx 


INTEGRAL  CALCULUS.  21 

Definite  Integrals. — On  evaluating  the  integral  \y  .  dx  .\t 
two  constant  quantities  a  and  h  are  separately  substituted  for  a;, 
and  the  former  result  substracted  from  the  latter,  the  difference 
is  termed  a  definite  integral  between  limits  a  and  h,  being  denoted 

y  .  dx.    Thus  {x^.dx  =  -j,  and  /    x^  dx  =  — j —  =  16i. 

The  following  definite  integrals  are  of  frequent  occurrence: — 
2 


cose  .dd  =  1 
0  0 


J      B'mO.de  =  1  J 

0 

0  0 

J  J  n       n-2  ■: 


cos^  a  .  cf0  =  7 
0  ^ 


I 


-  when 
00  ^ 

n  is  an  odd  integer. 

n-1    n-3  3    1    IT    , 

=  — ^  •  — 2     *    •    '4*2*2  ^^^°  *^  ^^  ^^  ®^'®"  integer. 


2 

sin  ^e  .  cos  "0  .  de  (m  and  n  being  integral) 


0 

_  hn-l){m-3)  .  ■  .  (n-Djn-S)  .   . 
~      {m  +  n)(in  +  n  —  2){m  +  n-i)  .  .  . 
n  is  odd. 


when  either  m  or 


(7;i-l)(m-3)  .  .  .  (n-l)(rt-3)  .   .   .      tt 
:       (,n^n)(m  +  n-2){m  +  n     4)   .  .   .       "  ¥  ^^"°   ^'^  ^"^ 
71  are  both  even. 


22 


GEOMETRY. 


PRACTICAL    GEOMETRY. 

1.  From  any  given  2Joint  in  a  straight  line 
to  erect  a  per  pen  dicv  lar.     (  Fig.  1 2 . ) 

On  each  side  of  the  point  A  in  the  line  from 
which  the  perpendicular  is  to  be  erected  set  off 
equal  distances  xb,  Ac ;  and  from  b  and  c  as 
centres,  with  any  radius  greater  than  Ab  or  Ac, 
describe  arcs  cutting  each  other  at  d,d';  a  line 
drawn  through  dd'  will  pass  through  the  point 
A,  and  Ad  will  be  perpendicular  to  be. 

2.  To  erect  a  perjfendicular  at  or  near  the 
end  of  a  line.    (Fig.  13.) 

With  any  convenient  radius,  and  at  any 
distance  from  the  given  line  ab,  describe  an 
arc,  as  BAG,  cutting  the  given  point  in  A ; 
through  the  centre  of  the  circle  N  draw  the 
line  BNC :  a  line  drawn  from  the  point  A, 
cutting  the  intersection  at  c,  will  be  the 
required  perpendicular, 

3.  To  divide  a  line  into  any  mimber  of  equal paHs. 
Let  AB  be  the   given   straight  ^la.  14. 

line  to  be  divided  into  a  number  of 
equal  parts ;  through  the  points 
A  and  B  draw  two  parallel  lines  Ac 
and  DB,  forming  any  convenient 
angle  with  ab  ;  upon  AC  and  DB  set 
off  the  number  of  equal  parts  re- 
quired,asA-1, 1-2, &c.,B-l, 1-2,  &c. ; 
join  A  and  d,  1  and  3,  2  and  2,  3 


J^-"/ 


S-""/ 


D 

and 


-jk:^^ 


1,  c  and  B,  cutting  AB 


in  a,  b,  and  c,  which  will  thus  be  divided  into  four  equal  parts. 

4.  To  find  the  length  vf  any  given  arc  of  a  circle.    (Fig.  15.) 

With  the  radius  Ac?,  equal  to  one- 
fourth  of  the  length  of  the  chord  of  the 
arc  AB,  and  from  A  as  a  centre,  cut  the 
arc  in  c  ;  also  from  b  as  a  centre  with 
the  same  radius  cut  the  chord  in  b ; 
draw  the  line  cb,  and  twice  the  length 
of  the  line  cb  is  the  length  of  the  arc  nearly. 

5.  'To  draw  from  or  to  the  cir- 
cumference of  a  circle  lines  tend- 
ing towards  the  centre,  7vhen  the 
centre  is  inaccessible.    (Fig,    16.) 

Divide  the  given  portion  of 
the  circumference  into  the 
desired  number  of  parts ;  then 
with  any  radius  greater  than  the 
distance  of  two  parts,  describe  arcs  cutting  each  other  as  Al,  cl 


GI'OMK'niT. 


Fig.  17. 


draw  the  lines  Bl,  c2,  etc. ,  which  will  lead  to  the  centre,  as  required. 
To  draw  the  end  lines  Ar',  Yr,  from  B  and  E  with  the  same  radii 
as  before  describe  the  arc  r' ,  r,  and  with  the  radius  Bl,  from  A  as 
centre,  cut  the  former  arcs  at  r',  r  ;  lines  then  drawn  from  Ar' 
and  Fr  will  tend  towards  the  centre,  as  required. 

6.  To  describe  an  arc  of  a  circle  of  large  radius.  (Fig.  17.) 
Let  A,B,  c  be  the  three  points  through 

whicli  the  arc  is  to  be  drawn  ;  join  ba 
and  BC  ;  then  construct  a  flat  trian- 
gular mould,  having  two  of  its  edges 
perfectly  straight  and  making  with 
each  other  an  angle  equal  to  ABC. 
Each  of  the  edges  should  be  a  little 
longer  than  the  chord  Ac.  In  the  points  A,  c  fix  pins  ;  and  fix  a 
pencil  to  the  mould  at  b,  and  move  the  mould  so  as  to  keep  its 
edges  touching  the  pins  at  A  and  c,  when  the  pencil  will  describe 
the  required  arc. 

7.  Another  method.    (Fig.  18.) 

Fig.  18.  Draw   the    chord    ADC,    and 

draw  EBF  parallel  to  it ;  bisect 
the  chord  in  D  and  draw  db  per- 
pendicular to  AC;  join  AB  and 
BC ;  draw  AE  perpendicular  to 
2     o    2    r~c  AB  and  CF  perpendicular  to  BC ; 

also  draw  A;i  and  c/i  perpendicular  to  AC ;  divide  AC  and  EF 
into  the  same  numbei'  of  equal  parts,  and  A/?,  en  into  half  that 
number  of  equal  parts  ;  join  1  and  1,  2  and  2,  also  B  and  s,  g, 
and  B,  and  t,  t ;  through  the  points  where  they  intersect 
describe  a  curve,  which  will  be  the  arc  required. 

8.  To  describe  an  ellipse,  the  major  and  minor  axes  being 
given.     (Fig.  19.) 

Let    AB    bo    the    major     and    CD    the 

minor    axis,     bisecting     each     other     at 

right  angles  in  tlie  centre  e  ;   from  c  as 

a  centre,  with  ea  as  radius,  describe  arcs 

cutting  AB  in  f  and  f',  which  will  be  the 

foci   of   the   ellipse ;    between   b   and   F 

set  off  any  number  of  points,  as  1,  2  (it 

is  advisable    that  these    points    should 

be  closer  as  they  approach  f). 

From  F  and  f',  with  radius  Bl,  describe  the  arcs  G,  G',  g",  g'". 

From  F  and  f',  with  radius  Al,  describe  the  arcs  H,  h',  h",  h'", 

intersecting  the  arcs  G,  g',  g",  G"'in  the  points  i,  i,  I,  i,  which  will 

be  four  points  in  the  curve. 

Then  strike  arcs  from  f,  f'  first  with  a2,  then  with  b2  ; 
these  radii  intersecting  will  give  four  more  points.  Proceed 
in  this  way  with  all  the  points  between  e  and  F;  the  curve  of 
the  ellipse  must  then  be  traced  through  these  points  by  hand. 


24 


GEOMETRY. 


Fig.  20.  9.  Another  method.     (Fig.  20.) 

Let  AB  and  cd  be  •  the  axes  ;  find 
F,  f',  the  two  foci  as  before  ;  join  CF, 
cf'  ;  make  an  endless  thread  equal  in 
length  to  the  perimeter  of  the  triangle 
cff',  and  passing  it  round  two  draw- 
ing-pins at  F  and  f',  draw  it  taut  by 
means  of  a  pencil-point  p,  so  as  to 
make  a  triangle  pff'  equal  in  peri- 
meter to  cff'  ;  move  the  pencil-point 

P  along,  keeping  the  thread  taut,  and  the  required  curve  will  be 

described. 

10.  Another  method.  (Fig.  21.)  (Ap- 
proximate.) 

At  0,  the  intersection  of  the  two  dia- 
meters, as  a  centre,  with  a  radius  equal 


—  to  the  difference  of  the  semi-diameters, 

describe  the  arc  ab,   and  from   J   as   a 

centre  with  half  the  chord  bca  describe 

the  arc  cd ;  from  o  as  centre  with  the 

distance  od  cut  the  diameters  in  dr,  dt ; 

draw  the  lines  rs,  rs,  ts,  ts,  then  from  r  and  t  describe  the  arcs 

SDS,  scs ;  also  from  d  and  d  describe  the  smaller  arcs  BAS,  SBS, 

which  will  complete  the  ellipse  required. 

This  method  is  applicable  when  the  minor  axis  is  at  least 
§  the  major. 

11.    To  draw  a   tangent   and  a   normal    to   an   ellipse   at 
any  point.     (Fig.  22.) 

Fig.  22.  Let  G  be  the  point;  from  f,  f',  the 

two  foci  of  the  ellipse,  draw  straight 
lines  through  G  and  produce  them  ; 
bisect  the  angle  made  by  the  pro- 
duced parts,  by  on,  then  gii  is  normal 
to  the  curve  ;  at  G  bisect  the  angle 
formed  by  fg  and  f'g  produced,  by 
15,  then  IJ  will  be  the  tangent  to  the 
curve  at  G. 


12. 

given. 


Fig. 23. 


To  desorihe  an  eJUptlo  arc,  the  sp'cn  and  height  betjig 
(Fig.   23.)      (Approximate.) 

Bisect  with  a  line  at  right  angles  the 
chord  or  span  AB  ;  erect  the  perpendicular 
AQ,  and  draw  the  line  QD  equal  and  parallel 
to  AC ;  bisect  AC  in  c,  and  aq  in  n  ;  make 
CL  equal  to  CD,  and  draw  the  line  lcq  ; 
draw  also  the  line  nsu,  and  bisect  SD  with 
a  line  KG  at  right  angles  to  it,  and  meeting 
the  line  LD  in  G  ;  draw  the  line  GKQ,  and 
make  cp  equal  to  CK,  and  draw  the  line 
G^2 ;  then  from  G  as  centre  with  the  radius 


GEOMKTIIT. 


25 


Fig.  24. 
I 


GD  flescribe  the  arc  sd2,  and  from  K  and  p  as  centres  with  the 
radiu3  ak  describe  the  arcs  as  and  2b,  which  complete  the  arc, 
as  required.  This  method  gives  good  results  for  cUipsev^  of 
all  proportions. 

13.  Another  method.     (Fig.  24.) 
Bisect  the  major  axis  ab,  and  fix  at 

right  angles  to  it  a  straight  guide,  as 
bo  ;  prepare  of  any  mat-crial  a  rod  or 
staff,  def  ;  at  /  fix  a  pencil  or  tracer  so 
that  df  is  one-half  ab,  and  at  E  fix 
a  pin  so  that  ef  is  one-half  the  minor 
axis  ;  move  the  st-aff,  keeping  its  end 
d  to  the  guide,  and  the  pin  e  to  ab, 
and  the  tracer  will  describe  a  half  of  the  arc  required. 

14.  To  describe  a  parabolic  aro  when  its  height  and  base 
are  given.     (Fig.  25.) 

Let  CD  be  the  base  and  ab  the  height;  set  them  off  as 
shown  in  the  figure,  so  that  cb  =  CD,  and  complete  the  rect- 
.angle  cdfe  ;  divide  EC  and  fd  into 
any  number  of  equal  parts,  say  three, 
at  a,  b,  c,  and  d;  join  Aa,  Ab,  ag, 
Ad ;  divide  ae,  af,  bc,  and  bd  into 
the  same  number  of  equal  parts  at 
e,  g,  k,  Jn,f,  h,  I,  n;  join  e/,  gh,  hi, 
mn,  cutting  Ab,  Aa,  Ac,  Ad  at  q,  p,  r, 
and  s.  A  line  drawn  through  c  q p  A  r  s  y>  will  i;e  the  parabola 
required. 

15.  Another  viethod,  .  when  the 
directrix  and  focus  are  given.  (Fig. 
26.) 

Place  a  straight-edge  to  the  direc- 
trix ab,  and  apply  to  it  a  square  cde  ; 
to  the  end  c  of  the  square  fasten  a 
thread,  and  pin  the  other  end  to  S  the 
focus,  making  the  length  of  the  thread 
equal  to  ce  ;  slide  the  square  along  the 
straight-edge,  holding  the  thread  taut 
against  the  edge  of  the  square  by  a 
pencil  P,  by  which  the  curve  is  de- 
scribed. 


Fio  25. 
A-        A        k 


1 

r 

w 

'^ 

5 

\ 

Fig.  26. 


IG.  To  describe  a  hyperbola,  the  diameter,  abscissa,  and  double 
ordinate  being  given.     (Fig.  27.) 

Let  AB  be  the  diameter,  BC  its  abscissa,  and  DE  its  double 
ordinate ;  then  through  b  draw  gf  parallel  and  equal  to  de  ; 
draw  also  DO  and  ef  parallel  to  the  abscissa  BC. 


26 


GEOMETRY. 


Divide  DC  and  CE  into  the  same  number  of  equal  parts,  as 
1,2,  &c.,  and  from  the  points  of  division  draw  lines  meeting  in  A. 

Divide  gd  and  ef  each  into  the  same  number  of  parts  as  DC 
or  c*E,  and  from  the  points  of  division  1',  2',  &c.,  draw  lines 
meeting  in  b. 


Fio    27. 


/ll'&'uV, _«=• 

I  \  \\\\      . 

/'//   '   !   !    1  \  \\\     ^JJ 


// 1    I    \    \    \    \    \\>. 

/    ;    I    I     I    \    \  \   i3 
/    /     I     I     \    \    \ 
/     /     I     I     \    V    \    \i 
i._(_  J__L_i  _\ \ 

6    3      S-     I      C      *     ^     a      E 


The  points  of  intersection  of  the  lines  1  and  1',  2  and  2',  &c., 
thus  found,  will  be  points  in  the  required  curve. 

17,  Another  method,  when  the  fool  and  a  point  on  the  curve 
are  given.     (Fig.  28.) 

A  hyperbola  is   a   curve   such   that  the   difference   of  the 
distances  of  any  point  in  the  curve  from  the  two  foci  is  equal  to 
the  transverse  axis ;  and  this  pro- 
perty  suggests   the  following  me- 
chanical construction : — 

Let  P  (fig.  28)  be  any  point  on 
the  curve,  and  F  and  F,  its  foci; 
join  FF,  and  produce  it,  making 
XX'  the  axis  ;  draw  pm  perpendicu- 
lar to  XX',  and  produce  it  to  Q, 
making  mq  equal  to  pm  ;  bisect  ff, 
at  c,  and  produce  PC,  QC,  to  CP', 
CQ',  making  them  equal  to  CP  or 
CQ.  The  p,  p',  Q,  q'  are  four  points  on  the  curve.  From  one  of 
them,  say  P,  stretch  two  pieces  of  string  pf  and  PF,,  fastening 
them  to  the  paper  at  F  and  F„  and  simply  knotting  them  at  P  ; 
slip  a  small  bead  over  them  at  P,  and  taking  hold  of  P  and 
keeping  the  thread  taut,  slide  the  bead  along  the  threads,  and 
the  bead  will  describe  the  curve  as  far  as  the  axis.  Kepeat  this 
process  at  p',  q,  and  q'. 


GEOMETRY. 


27 


18.     To    describe    a    rectangular 
asympfoles  and  a  'point  on  the  curve. 


Let  ox,  OY  be  the 
asymptotes,  and  P  the 
given  point.  Draw  rar 
parallel  to  OY,  and  Ps 
parallel  to  ox ;  set  off  any 
ordinates(generally  equi- 
distant for  convenience) 
11,22,33,  44,55,  66,  and 
join  0  to  the  intersections 
of  these  ordinatea  with 
PS,  cutting  PM  at  1',  2' 


hyperbola,    given    tha 
(Fig.  29.) 

Fig.  23. 


6 5^ 


B'',  etc.;  through  1'  draw  I'l  parallel  to  ox,  cufctinj  11  in 
through  2',  2'ii  cutting  22  in  ii,  and  so  on  for  m,  iv,  v,  and  vi ; 
then  P,  I,  II,  III,  etc.,  are  points  on  the  required  curve. 

19.  Given  five  points  on  any  conic  to  obtain  any  number  of 
additional  points  desired.     (Fig.  30.) 

Denote  the  given  points  by  a,  b,  c,  d,  e.  Draw  any  line 
pap'  through  A,  on  which  it  is  required  to  find  a  sixth  point. 

Fig.  30. 


Join  AB,  DE,  cutting  at  x,  and  CD  cutting  pap'  at  Y.  Join  BC, 
cutting  XY  in  z;  join  EZ,  cutting  pap'  in  F.  f  is  the  required 
point  on  the  curve  ;  by  drawing  additional  linos  through  A, 
any  number  of  points  on  the  curve  may  be  obtained., 

20.  To  construct  a  catenary  approximately.     (Fig.  30a.) 
Let  E  be  the  lowest  point  in  the  curve,  oe  its  parameter, 
and  ox  its  directrix.    Make  ae  ecjual  to  OE;   then  with  a  as 


28 


GEOMETRY. 


centre  and  ae  as  radius  describe  the  small  arc  ef.  Join  fa 
and  produce  it  to  M  and  to  b,  makin*  bf  equal  to  FM;  then 
with  B  as  centre  and  bf  as  radius  describe  the  small  arc  fg. 

Fia.  80a. 


Join  BO  and  produce  it  to  N  and  to  c,  making  CG  equal  to 
GN  ;  then  with  c  as  centre  and  CG  as  radius  describe  the  small 
arc  Gn.  Proceed  in  a  similar  manner  till  the  curve  is  of  the 
required  length. 


21.   To  obtain  hy  measurement   the  length  of  any  direct 
line,  though  intercepted  by  some  material  object.    (Fig.  31.)     , 

Suppose  the  distance  between  ^^^   3^- 

A  and  B  is  required,  but  the 
straight  line  is  intercepted  by 
the  object  G.  On  the  point  d, 
with  any  convenient  radius, 
describe  the  arc  cc',  and  make 
the  arc  twice  the  radius  dc  in 
length  ;  through  c'  draw  the 
line  dc'e,  and  on  e  describe  another  arc  ff  equal  in  length  to 
the  radius  dc  ;  draw  the  line  efr  equal  to  efd  ;  from  r  describe 
the  arc  g'g,  equal  in  length  to  twice  tho  radius  rg\  continue  the 
line  through  rg  io  m  '.  then  A  and  B  will  make  a  right  line, 
and  de  or  er  will  equal  the  distance  between  dr,  by  which  the 
distance  between  ab  is  obtained,  as  required. 


22.    To   ascertain    the    distance   geometrically    of   an    in- 
accessible object  on  a  level  plane.     (Fig.  32.) 

Let   it   be  required  to  find   the   distance   between   a   and 
c,    A  being  inaccessible.     Produce   ab   to  any   point   d,   and 


GEOMETKY. 


29 


bisect    «T>    m    C  ;    through    d    draw  Fio.  82. 

Da,  making  any  angle  with  DA,  and 
take  DC  and  db  respectively  and  set 
them  off  on  Dor  as  D 6  and  do  ;  join 
Be,  cb,  and  a6  ;  through  e,  the  inter- 
section of  Be  and  cb,  draw  def  meet- 
ing a6  in  F  ;  join  bf  and  produce  it 
till  it  meets  Da  in  a  :  then  ab  will  be 
equal  to  ab,  the  distance  required. 

23.  Another  method.     (Fig.  33.)  Fig. 

Produce  ab  to  any  point  d  ;  draw  the 
line  T)d  at  any  angle  to  the  line  ab  ;  bisect 
tlie  line  Dd  in  C,  through  which  draw  the 
line  B&,  and  make  Cb  equal  to  Bc;  join  AC 
and  db,  and  produce  them  till  they  meet 
at  a  :  then  ba  will  equal  ba,  the  distance 
required. 

24.  To  measure  the  distance  between  Uvo 
objects,  both  being  inaccessible.    (Fig.  34.) 

Let  it  be  required  to  find   the  distance  ^iq.  34. 

between  the  points  a  and  B,  both  being  in- 
accessible. From  any  point  c  draw  any  line 
Cc,  and  bisect  it  in  d  ;  produce  ac  and  Be, 
and  prolong  them  to  E  and  F  ;  take  the  point 
E  in  the  prolongation  of  Ac,  and  draw  the» 
line  EDe,  making  De  equal  to  de. 

In  like  manner  take  the  point  F  in  the 
prolongation  of  Be,  and  make  d/  equal  to  df  ; 
produce  ad  and  ec  till  they  meet  in  a,  and 
also  produce  bd  and  /c  till  they  meet  in  b: 
then  the  distance  between  the  points  a  and 
b  equals  the  distance  between  the  inaccessible 
points  a  and  B. 

25.  To  cut  a  beam  of  the  strongest  section 
from  any  round  piece  of  timber.    (Fig.   35.) 

Divide  any  diameter  CB  of  the  circle  into 
three  equal  parts ;  from  d  or  e,  the  two  points 
of  division  in  CB,  erect  a  perpendicular  cutting 
the  circumference  of  the  circle  in  d  or  a  ; 
draw  CD  aad  db,  also  ac  equal  to  db  and  ab 
equal  to  CD  :  the  rectangle  abcd  will  be  the 
section  of  the  beam  required. 

Note. — To    get    the    stiff  est    beam    make    cd  =  ^    cb    and 
proceed  as  before. 


FlQ. 


GEOMETRY. 


26.  To  describe  the  proper  form 
of  a  flat  plate  by  which  to  construct 
any  given  frustum  of  a  cone.  (Fig.  36.) 

Let  ABCD  represent  the  required 
frustum  of  a  cone  ;  continue  the  lines 
AC  and  BD  till  they  meet  in  E  ;  from  E 
b-s  a  centre,  with  ed  as  radius,  describe 
the  arc  dii,  and  from  the  same  centre, 
fwith  EB  as  radius,  describe  the  arc  Bi; 
make  Bi  equal  in  length  to  twice  agb, 
equal  to  the  circumference  of  the  base 
of  the  cone  ;  draw  the  line  ei  :  then 
BDHI  is  the  form  of  the  plate  required. 


27.  To  find  the  development  of  the  frustum  of  a  right  cone 
tchen  cut  by  an  angle  inclined  to  the  base.     (Fig.  37.) 

Let  ABCD  represent  the  required 
frustum  of  the  cone  ;  continue  the  lines 
AC  and  BD  till  they  meet  at  E  ;  divide 
the  circumference  of  the  base  of  the 
cone  into  any  number  of  equal  parts- 
say  12 — in  the  points  1,  2,  3,  etc.  ; 
join  the  projections  of  these  points  to 
E  ;  next  find  the  development  of  the 
Is  base  of  the  cone,  as  shown  in  the  pre- 
ceding example,  and  on  it  set  off  the 
same  number  of  points — viz.  12 — and 
draw  lines  from  them  to  e  ;  project 
the  points  of  intersection  of  each  of  the 
lines  El,  e2,  e3,  etc.,  with  the  line  CD, 
horizontally  on  to  either  of  the  slant 
sides  (say  eb)  ;  then  from  e  as  centre 
measure  the  distance  down  along  eb  to 
the  foot  of  each  projection  and  set  it 
off  on  the  corresponding  numbers 
(measuring  from  e)  in  the  develop- 
ment :     a    line    drawn    through    these 

points    will    give    the    curve   of    the    top    of    the   section,    as 

required. 

28.  To  find  the  development  of  the  frustum  of  a  cylinder 
when  cut  by  a  plane  inclined  to  the  base.     (Fig.  38.) 

Let  ABCD  represent  the  required  frustum  of  a  cylinder  ; 
divide   the  base  into  any   number   of  equal   parts — say    12 — 


GEOMETRY. 


81 


and  draw  lines  through  those  points  on  the  cylinder  parallel 
to  AC  and  BD  ;  draw  a  line  efg  equal  in  length  to  the  circum- 
ftr.rence  of  tlie  cylinder,  and  divide  it  into  the  same  number 
;>ff  j^u'ts  ;  on  each  point  of  division  set  up  perpendiculars  to 

Fig.  ns. 


it,  making  eh  and  GK  equal  in  length  to  bd,,  and  make  Fi  equal 
in  length  to  AC;  then  take  the  height  at  1  and  set  it  up  on 
the  corresponding  number  on  each  side  of  Fi,  and  so  on  with 
each  number  :  a  line  ti*accd  through  the  points  thus  obtained 
will  be  the  curve  of  the  required  development. 


29.  To   find   the  approximate  development   of  any   given 
portion   of  a  segment  of  a 


(Figs.    39,   40,   and 


sphere. 
41.) 


Let  ABC  (fig.  39)  be  the 
middle  section  of  the  seg- 
ment, and  CFG  in  the  plan 
(fig.  40)  the  portion  to  be 
developed;  bisect  ab  (fig.  39) 
in  E,  and  set  up  the  perpen- 
dicular EC  ;  divide  the  arc 
AC  into  any  given  number 
of  equal  parts — say,  four — 
and  through  the  points  of 
division  draw  the  lines  1  1, 
2  2,  etc.,  parallel  to  ab  ;  on 
the   plan    (fig.    40)    from    c 


Pig.  40. 


as  a  centre,  with  the  radius  1  1  taken  from  fig.  39,  draw  the 
arcs  1  1  cutting  fc  and  CG  in  1  and  1,  and  so  on  with  2  2  and 
3  3;  draw  any  line  bc  (fig.  41),  making  it  equal  in  length  to 
bc  (fig.  39),  and  on  it  set  off  the  same  number  of  equal  parts; 
at  each  point  of  division  draw  linos  perpendicular  to  bc,  and 
number  them  the  same  as  on  fig.  39. 


82 


GEOMETRY. 


Measure  the  length  of  the  arc  1  1  in  fig.  40,  and  set  off 
half  of  it  on  each  side  of  co  on  line  1  1,  and  so  on  with  each 
arc,  including  FG  ;  a  line  traced  through  the  points  thus 
obtained  will  give  the  curve  of  the  sides  of  the  given  portion 
of  the  segment  when  it  is  developed.  To  describe  the  curve  at 
the  bottom  of  the  figure,  take  one-fourth  of  the  circumference 
of  the  base  as  a  radius,  and  from  F  and  o  as  centres  describe 
arcs  cutting  bo  in  s  ;  then  from  s  as  centre,  with  the  same 
radius,  describe  the  arc  fbg,  which  w^U  bo  the  curve  of  the 
bottom  of  the  figure,  as  required. 

Should  the  top  of  the  figure  be  cut  off  at  the  lino  1  1 
(fig.  39),  from  s  as  a  centre  in  fig.  41  describe  the  arc  1h1, 
which  will  be  the  curve  of  the  top  of  the  figure,  as  required. 


30.   To   -find  the  a'p'proxlmate  development  of  any  given 
'portion  of  a  paraboloid.     (Figs.  42,  43,  and  44.) 


Fig.  42. 


The  development  is  found 
in  the  same  manner  as  that 
of  a  portion  of  a  segment 
of  a  sphere,  as  described  in 
the  last  example  (No.  29), 
with  but  one  exception — 
that  is,  the  length  of  the 
radius  for  describing  the 
bottom  curve  of  the  figure, 
which  instead  of  being  equal 
to  one-fourth  of  the  circum- 
ference, as  in  example 
No.  29,  is  equal  to  one-half 
the  length  of  the  arc  acb 
(fig.  42)  in  this  example. 


31.  To  find  the  development  of  an  entablature  plale. 

Let  fig.  45  be  the  side  elevation,  fig.  46  the  front  elevation, 
fig.  47  the  plan,  and  fig.  48  the  development  of  the  figure  ; 
divide  ado  (fig.  46)  into  eight  equal  parts,  and  from  the 
points  of  intersection  draw  lines  parallel  to  abc,  cutting  CD 
(fig.  45)  in  the  points  1,  2,  etc.;  on  bd  (fig.  45)  erect  a  perpen- 
dicular EC,  and  from  the  points  on  CD  draw  lines  parallel  to 
BED.  From  fig.  46  take  the  points  1,  2,  etc.,  on  abc  and  set 
them  off  on  afc  (fig.  47),  and  erect  perpendiculars  from  afc  at 
these  points.  From  c  (fig.  45)  along  ce  measure  the  points  c,  1, 
c,  2,  etc.,  and  set  them  off  on  their  corresponding  lines  from 
afc  in  fig.  47;  draw  a  line  through  those  points,  then  measure 
it  with  its  divisions  and  set  it  off  in  fig.  48  as  a  straight  line 


GEOMETRY.  §3 

AEC,  and  at  the  points  of  division  erect  perpendiculars,  con- 
tinuing them  either  side  of  the  line  akc  ;  measure  tiie  distances? 
1,  1;  2,  2,  etc.  (fig.  45),  on  either  side  of  ce,  and  set  them  olf 


Fig.  45 


from  AEC  (fig.  48)  on  their  corresponding  lines,  and  on  their 
respective  sides  of  aec.    These  will  give  the  development. 

32.  To  describe  a  cycloid,  the  generating  circle  being  given. 
(Fig.  49.) 

Let  b6  be  the  generating  circle  ;  draw  a  line  abc,  equal  to 
the  circumference  of  the  generating  circle,  by  dividing  the 
circle  into  any  number  of  given  parts,  as  1,  2,  3,  eta.,  and 


setting  off  half  that  number  of  parts  on  each  side  of  B;  draw 
lines  from  the  intersections  of  the  circle  1,  2,  3,  etc.,  7,  8,  9, 


34 


GEOMETRY 


etc.,  parallel  to  AC  ;  set  olf  one  division  of  the  circle  outwards 
on  the  first  lines  5  and  7,  two  divisions  on  the  next  lines  4 
and  8,  then  three  on  the  next,  and  so  on  :  then  the  intersection 
of  those  points  on  the  lines  1,  2,  3,  etc.,  will  be  points  in  the 
curve. 


33.  To  draw  a  trochoid  or  wave-form,  the  height  and  length 
being  given.     (Fig.  50.) 

Draw  AB  equal  to  the  length  ;  with  centre  C  on  ab  produced 
describe  a  circle  whose  diameter  is  equal  to  the  height. 
Divide  the  circumference  into  a  convenient  number  (say  12) 
of  equal  parts  0,  1,  2,  3,  .  .  .  ,  CO  being  vertical.  Diride  AB 
into  the  same  number  of  equal  parts  f,  o,  H,  .  .  .  From 
A,  F,  G,  .  .  .  B,  draw  Aa,  f/,  0*7,  .  .  ,  b6,  parallel  and  equal 
c  0,  c  1,  c  2,  .  .  .  C  0,  respectively.  A  curve  drawn  through 
the  points  a,  f,  g,  .  .  .  b  \%  the  required  trochoid. 

Fig.  50. 


liLH 


*,    F   6 


i^.  To  describe  an  epicycloid,  the  generating  circle  and  the 
directing  circle  being  given.      (Fig.   51.) 


Let  bd  be  the  generating  circle,  and  ab  the  directing  circle; 
divide  the  generating  circle  into  any  number  of  equal  parts 
(say  10)  as  1,  2,  3,  etc.,  and  set  off  the  same  distances  round 
the  directing  circle;  draw  radial  lines  from  a  through  these  last 
points,  and  produce  them  to  an  arc  drawn  with  a  as  centre  and 
AE  as  radius,  as  shown  by  cccc  and  c'c'c'c'  on  the  diagram  ;  draw 


GEOMETRY. 


35 


concentric  arcs  also  through  all  the  points  on  the  generating 
circle,  with  A  as  centre ;  then  taking  c,  c,  c,  c  and  c',  c',  c',  c'  as 
centres,  and  be  as  radius,  describe  ai'cs  cutting  the  concentric 
circles  at  1',  2',  etc.  :  the  points  thus  found  will  be  points  in  the 
required  curve. 


35.  To  describe  a  hypo- 
cyrloid,  the  generatino 
circle  and  the  directin(^ 
circle  being  given,  (Fig. 
52.) 

Proceed  as  in  the  epi- 
<5ycloid,  the  exception 
being  that  the  construction 
lines  are  drawn  within  the 
directing  circle  instead  of 
outside,  as  in  the  epicy- 
cloid. 


FiO.  52. 


^^ 


36.  To  describe  the  involute  of  a  circle.     (Fig.  53.) 

Let  AD  be  the  given  circle,  which  divide  into  any  equal 
number  of  parts  (say  12)  as  1,  2,  3,  etc.  ;  from  the  centre 
draw  radii  to  these  points  ;  then  draw  lines  (tangents)  at  right 
angles  to  these  radii.  On  the  tangent  to  radius  No.  1  set  off 
from  the  circle  a  distance  equal  to  one  part,  and  on  each  of  the 

Fig.  53. 


tangents  set  off  the  number  of  parts  corresponding  to  the 
number  of  its  radius,  so  that  No.  12  has  twelve  divisions  set 
off  on  it  (that  is,  equal  to  the  circumference  of  the  circle)  : 
a  line  traced  through  the  ends  of  these  lines  will  be  the  curve 
required. 


36 


MENSURATION    OF    AREAS    AND    PERIMETERS. 


37.  To  -find  the  dip  of  the  horizon.     (Fig.  o3a.) 

Let  0  denote  the  centre  of  the  earth,  pb  a 
tang-ent  from  the  eye  of  an  observer  looking 
from  a  height  ap  to  the  earth's  surface  at  B; 
then  B  is  a  point  on  the  horizon:  draw  PC  at 
right  angles  to  po  ;  then  the  angle  BPC  is 
called  the  dip  of  the  horizon. 

Let  OP  cut  the  earth's  surface  at  a,  and  let 
the  angle  bpc  be  denoted  by  Q  ;  with  distances  in  miles, 
PB  =    V  2  .  AP  .  AG  approximately  =    V  8,000  x  AP  ;    and  Q  in 
degrees  =  1-28  Vap. 

MENSUEATION. 
I.  Mensuration  of  Areas  and  Perimeters. 
1.  To  find  the  area  of  miy  ^parallelogram.     (Fig.  54.) 

EuLE. — Multiply    the    length    by 

the    perpendicular    height,    and    the 

~'Tt     product  will  be  the  area.     Thus,  if 

/ij     A  =  the    area,    a  =  the   length,    and 

'   i^     b  =  the  perpendicular  height,  tlien 

A  =  ah. 


Fig. 54. 


*     L 


2.  2'o  find  the  area  of  a  trapezoid.    (Fig.  55.) 


Fig.  55. 


...& 


Rule. — Multiply  the  sum  of  the  parallel 
sides  by  the  perpendicular  distance  between 
them;  half  tha  product  will  be  the  area.  Thus, 
if  A  =  the  area,  b  and  a  =  the  parallel  sides, 
and    c  =  the    perpendicular    distance    between 

tJiem,    then    a  =.  (^±_&)£ 
'  2 

3.  To  find  the  area  bf  any  triangle.      (Fig.   56.) 

Pjq  gg  EuLE. — Multiply  the  base  by  the  per- 

pendicular height  ;  half  the  product  will 
be  the  area.  Thus,  if  a  =  the  area,  b  =  the 
base,   and   d  =  the    perpendicular    height, 

O-.  —  TT^       then  A  =  — - 

4.  Or,  if  the  lengths  of  the  3  sides  a,  h,  and  c  are  given,  then 
=  \/&{s-  a)  (5  -  6)  (s  -  c)  where  2s  =  a  +  6  +  c. 

5.  To  find  the  area  of  any  regular  'polygon.     (Fig.  56a.) 
Rule. — Multiply  the  sum  of  its  sides   by  a 

perpendicular  drawn  from  the  centre  of  the 
polygon  to  one  of  its  sides  ;  half  the  product  will 
be  the  area.  Thus  if  A  =  the  area,  c  =  the  number 
of  sides,  6  =  the  length  of  one  side,  and  a  =  the 

perpendicular,  then  A  =  -r- 


FiG.  56a. 


MKNSUUATIOX    OF    ARKAS    AND    PERIMETERS. 


37 


Table  of  Kegular  Polygons. 

A  =  the  angle  contained  between  any  two  sides. 

R  =  the  radius  of  the  circumscribed  circle. 

r  =  the  radius  of  the  inscribed  circle. 

s  =  the  side  of  the  polygon. 

5s 

^2           Name 

23M 

A 

R  =  SX 

r=sx 

S  =  RX 

s  =  rx 

Area=8^ 

.3   Trigon 

60° 

•57735 

•28868'l-73205'3-46410'     -43301 

4  1  Tetragon     . 

90° 

•70711 

-50000  1-41421  2-00000   1-00000 

5   Pentagon    . 

108° 

•85065 

-68819  1-175.57  1-45309    1-72048 

6   Hexagon     . 

120° 

1-00000 

•86603  1-000001-15470   2-59808 

7  1  Heptagon   . 

128|°  ;i-1.52.38'l-0.S826    -86777,  -96.315    3-63.391 

8  j  Octagon      . 

13.5°     1-306.56:1-20711    -76537!  -82843   4-82843 

i) 

Nonagon     . 

140°    1-46190 1-37374   -684041  -72794   6-18182 

10 

Decagon 

144°     1-61803 1-.5.3884   -61803   -64984   7-69421 

11 

Undecagon 

147fi°  1-77473  1-702841  -56347   -58725   9-,36,564 

12  1  Duodecagon 

150^     1-93185  1-866031-51764    -5359011-19615 

6.  To  find  the  area  of  a  quadrilateral. 

Rule. — Multiply  the  diagonal  d  by  the 
sum  of  the  two  perpendiculars  a  and  h  let 
fail  upon  it  from  the  opposite  angles;  half 
the  product  will  be  the  area.  Thus  if  A  = 
the  area,  a  and  J  =  the  perpendiculars,  and 
<Z  =  the  diagonal,  then 

.      (»  +  h)  d 


(l^^ig.  57.) 
Fio.  57. 


7.  To  find  the  cArcumference  of  a  circle,  the  diameter  being 
given  ;  or  to  find  the  diameter  of  a  circle,  the  circumference  being 
given. 

liULE.— Multiply  the  diameter  by  .3-1416,  the  product  will 
be  the  circumference ;  or  divide  the  circumference  by  3-1416, 
the  quotient  will  be  the  diameter. 


8.    To  find  the  length  of  any  arc  of  a  circle 


Rule  (I). — From  eight  times  the  chord 
of  half  the  arc  subtract  the  chord  of  the 
whole  arc  ;  one-third  of  tlie  remainder  will 
be  the  length  of  tlie  arc,  nearly.  Thus  if 
L  =:  length  of  the  arc,  c  =  chord  of  the 
whole  are,  c  =  chord  of  half  the  arc,  then 

He  -C 
^=       -6- 


38  MENSURATION    OF    AREAS    AND    PERIMETERS. 

Rule  (II). — The  radius  being  known,  multiply  together 
the  number  of  degrees  in  the  arc,  the  radius,  and  the  number 
•01745  ;  the  product  -will  be  the  length  of  the  arc.  Thus  if 
L  =  length  of  the  -arc,  d  =  degrees  in  the  arc,  R  =  radius, 
then. 

L  =  D  X  R  X  -01745. 

EuLE  (III). — (Applicable  to  any  fairly  flat  curve.')  Add 
to  the  chord  eight-thirds  the  square  of  the  maximum  height 
(or  versed  sine)  divided  by  the  chord.  The  sum  is  the  length 
of   the   curve,    very    nearly.      Thus    if    c  =  chord,    and    v  == 

8  V^ 

greatest  height  of  arc  above  chord,  length  =  c  -f-  -  — 

o    0 

9.  To  -find  the  diameter  of  a  circle,  the 
chord  and  versed  sine  being  given. 
(Fig.  59.) 

Rule. — Divide  the  square  of  half  the 
chord  by  the  versed  sine,  to  the  quotient 
add  the  versed  sine,  and  the  sum  vi'ill  be 
the  diameter.  Thus  if  D  =  the  diameter, 
c  =  the  chord,  and  v  =  the  versed  sine, 
then 


Fig. C9. 


Af----C 


(iy 


+  v 


Fig.  CO. 


I 


10.  To  find  the  length  of  any  ordinate  of  a  segment  of  a 
circle.      (Fig.   60.) 

Rule. — Find  the  radius  of  the  arc  of 
the  segment  (if  not  given)  by  the  pre- 
ceding formula;  and  from  the  square  root 
of  the  difference  of  the  squares  of  the 
radius  and  distance  of  the  ordinate  from 
the  centre  of  the  segment,  subtract  the 
radius  ;  and  to  the  result  add  the  height  of  the  segment,  and 
the  sum  will  be  the  required  ordinate.  Thus  if  ii  =  the  radius, 
X  =  the  distance  of  the  ordinate  from  the  centre  of  the 
segment,  v  =  the  height  of  the  segment,  and  Y  =  the  required 


ordinate,    then 


y  =  VR' 


R  +  V. 


11.  To  find  the  area  of  a  circle. 

Rule  (I). — Multiply  the  square  of  the  diameter  by  "7854, 
and  the  product  v/ill  equal  the  area,  nearly.  Thus  if  a  =  the 
area,  d  =  the  diameter,  then  a  =  D^  X  '7854. 

Rule  (II). — Multiply  the  square  of  the  circumference  by 
•07958,  and  the  product  will  be  the  area.  Thus  if  a  =  area, 
c  ^=  circumference,  then  a  =  c^  X  '07958. 


MENSURATION   OF    SUPERFICIES.  39 


Table  of  Properties  of  the  Circle.                    1 

TT  =3-14159265358979323846 

\/2  =1-41421356237300504880 

I  =1-57079632679489661923 
t 

n/^  =   -70710678118654752440 

2N/ir  =3-54490770181103205460 

4-  =  -78539816339744830962  2    /r  ^,.12837916709551257390 

IT                                                                                             ^^^       "^ 

i;  =  -52359877559829887308  J  yi  =   •886226925462758013C5 

Ts/2  =4-44288293816836624702i   /i=  •07032369794346953687 
's/J  =2-221441469079183I2351l  ^    ' 
Vi  =  1-77245385090351602730'       %  =  6-2831853071796,8647693 

y  I  =   •56418958354775628696.'         ;,  =   -63661977236768134308 

ifo  =  -«^^«                               V-«^-^ 

I  =  -3183                                           1^  =  9-5493 

ir2  =  9-870 

In  the  following  forraulie  A  =  area,  C  =  circumference,  D  =  diameter, 

s  =  side  of  square. 

Circumference                        ■=Dx7r-iix2jr=  x/a  \  2  V  tt 

Diameter                                 ^  c  x  -  =  v^A  x  2^/  - 

Radius                                       "  ^  ""  ^^;r  ^  ^/  -^  >^  s/ ^ 

Area                                         ^  r2  yjr  =  d'  x  -  =  i  j^n 

4 

Side  of  equal  square               =r  x  \/7r  =  D  x  ^v/7r  =  C  x  ^^'    - 

Hide  of  inscribed  square       =Dx  \/^  =  cx  -\/l=  ^/ A  x  ,^/  - 

Diameter  of  equal  circle       =  s  x  2^  - 

Diameter  of  circumscribing  circle  =  s  x  \/2 

Circumference  of  circumscribing  circle -s  x  ■jr\/2 

Circumference  of  equal  circle  =  s  x  2\/ir 

2 
Area  of  inscribed  square       =  A  x  - 

40         MENSURATION    OF    AREAS    AND    PERIMETERS. 

12.  To  find  the  area  of  a  sector  of  a  circle. 

Rule  (I.) — Multiply  the  length  of  the  arc  by  the  radius  of  the 
sector,  and  half  the  product  will  equal  the  area. 
A  =  area  of  sector,  l  =  length  of  arc,  R  =  radius. 

Rule  (II). — Multiply  the  number  of  degrees  in  the  arc 
by  the  area  of  the  circle,  and  g^^of  the  product  will  equal 
the  area.  Thus,  if  a  =  area,  D  =  number  of  degrees  in  the  arc 
a  =  area  of  circle,  then 

_  J^ 

~  B60 


13.   To  f.nd  the  area  of  the  segment  of  a  circle. 

Rule  (I). — Find  the  area  of  a  sector  having  the  same  arc 
as  the  segment  ;  then  deduct  the  area  of  the  triangle  con- 
tained betv.-een  the  chord  of  the  segment  and  the  radii  of  the- 
sector.  The  remainder  will  be  the  area  of  the  segment.  Thus, 
if  A  =  the  area  of  the  segment,  c  =  the  choi-d,  and  H  =  the 
height,  then 

Rule  (II). — To  two-thirds  of  the  product  of  the  chord  and 
height  of  the  segment,  add  the  cube  of  the  height  divided  by 
twice  the  chord  ;  the  sum  will  be  the  area  of  the  segment^ 
nearly.    Thus, 

/2cH   ,    H^\ 


Pj(5  gi  14.   To  find  the  area  of  a  circular  zone. 

^,..'>,;  (Fig.  61.) 

^    "^N  Rule. — Find   the  area  of   the   circle   of 

\  which  the  zone  forms  a  part,  and  from  it 

\  subtract  the  sum  of  the  two  segments  of  the 

A  J  circle  formed  by  the  zone  ;   the  remainder 

I  will  be  the  area.     Thus,  if  \  —  area  of  the 

V  /   zone,  a  and  b  =  the  area  of  the  two  seg- 

^      ^y      ments    respectively,    and    c  =  area    of    the 
circle,  then  \  =  c  —  {a  -\-  h). 


15.    To  find  the  ar&a  of  a  flat  circular  ring.     (Fig.  62.) 

Rule. — Multiply     the    sum    of    the    inside    and    outside 
diameters    by    their    difference,    and    the    result    by    "7854  ; 


MENSURATION    OF    AREAS    AND    PERIMETERS.  41 

Fra.  62. 

the  product  last  obtained  will  be  the  area. 
Thus,  if  A  =  area  of  ringf,  i)  =  diameter  of 
large  circle,  and  d  =  diameter  of  small 
circle,  then 

A=  •7854{{D  +  d)  {D-d)} 


16.  To  find  the  area  of  an  ellipse.     (Fig.  63.) 


Rule. — Multiply    together    the    trans- 
verse and  conjugate  diameters  of  the  ellipse, 

and  the  result  by  -  or  -7854 ;  the  product 

will  be  the  area.      Thus,  if  A  =  area  of 
ellipse,    a  =  the   conjugate  diameter,   and 
6  =  the  transverse  diameter,  then 
A  =  -ab  X  -7854. 


Fig.  63. 


17.  To  find  the  area  bounded  by  a  rectangular  hyperbola, 
two  ordinates  and  the  base.     (Fig.  8,  p.  15.) 

Rule. — Multiply  the  product  of  either  ordinate  and  the  corre- 
sponding abscissa  by  the  hyperbolic  logarithm  of  the  ratio  between 
the  two  abscissae.  Thus  the  area  of  ABCD  is  equal  to  ab  x  OB 
00 


lo; 


OB 


18.  To  find  the  area  bounded  by  a  cycloid  and  the  line 
joining  the  cusps.     (Fig.  11,  p.  18.) 

Rule. — Multiply  the  area  of  its  generating  circle  by  3 ;  or 
multiply  the  product  of  its  length  and  height  by  f . 

19.  To  find  the  area  bounded  by  a  trochoid  and  a  line 
joining  the  crests.     (Fig.  50,  p.   34.) 

Rule. — If  r  be  the  radius  of  the  rolling  circle  (or  the  length 
divided  by  2ir),  and  r  the  radius  of  the  tracing  circle  (or  one-half 
the  height  from  crest  to  trough),  the  required  area  is  equal  to  Trr 
(2r  +r). 

Note. — The  area  between  the  curve  and  the  line  joining  the 
troughs  is  vr  {2R-r). 

20.  To  find  the  area  of  a  segment  of  a  parabola. 
Rule. — Multiply  the  base  by  §  of  the  maximum  height. 


42 


MENSURATION    OF    AREAS    AND    PERIMETERS. 


21.  To  find  a  general  expression  for  the  area  of  any  plane 
curve. 

Using  cartesian  co-ordinates,  the  area  intercepted  between  the 
curve,  the  x  axis,  and  two  ordinates  distant  a  and  6  from  the  origin, 

'6 
is  equal  to  the  definite  integral    /    ?/ .  dx. 


J   a 


Using  polar  co-ordinates,    the  area  intercepted  between  the 
curve  and  two  radial  lines  making  angles  a  and  fi  with  ox,  is 

equal 


to  I    fr^  .  d6. 
J    o. 


Remark.— A.  curve  whose  equation  is  given  by 

y  =  a  +  bx  +  cx^  +  dx^  +  .  .  .  Ka;" 

is  said  to  be  a  parabolic  curve  of  the  n*^  order.  Thus  a  parabolic 
of  the  first  order  is  a  straight  line  ;  of  the  second  order  a  common 
parabola.  Eules  for  the  area  of  a  parabola  of  any  order  are 
applicable  also  to  curves  of  a  lower  order,  but  not  in  general  to 
those  of  a  higher  order. 

22.  To  find  the  area  of  a  parabola  of  the  third  order  when 
three  ordinates  are  given.     (Fig.  64.) 

Rule. — To  the  sum  of  the  two  endmost 
ordinates  add  four  times  the  intermediate 
ordinate  ;  multiply  the  final  sum  by  J  of  the 
common  interval  between  the  ordinates.  The 
^■^  result  will  be  the  area.  Thus,  if  2/1,  Vi,  and  1/3 
be  the  ordinates.  Ax  the  common  interval,  and 
X  the  area,  then 


Fig.  64. 


\ydA 
\yd 


'x=-f{yi  +  ^y-2+yi)' 


Note. — This  is  termed  Simpson's  first  rule, 

23.  To  find  the  area  of  a  parabola  of  the  third  order  when 
four  ordinates  are  given. 

Rule. — To  the  sum  of  the  two  end- 
most  ordinates  add  three  times  the 
intermediate  ordinates  ;  multiply  the 
final  sum  by  §  of  the  common  interval 
K^  between  the  ordinates :  the  result  will  be 
the  area.  Thus,  if  \ydx  =  ih.e  area,  then 
SAa!/ 


Fig. 65 

y 

y^ 

y-? 

AJB 

A.r 

AJC 

\ydx-- 


8 


■(Z/l  +  3l/2  +  3?/3-i-2/4). 


Note. — This  is  termed  Simpson's  second  rule. 


MENSURATION    OF    AREAS    AND    PERIMETERS.  43 


Table  showing  the  Multipliers   for  the  foregoing 

AND   SOME  other  RuLES, 
yu  yi,  ys,  etc.  =  the   ordinates,  and  Ax  =  the   common 
interval  or  abscissa  between  the  ordinates. 


1.  Trapezoidal  rule. 
Area  =  -—[yi  +  y^). 

2.  Rule  lor  parabola  of  the  third  order  with  three  ordinates. 

Area  =  -^ (t/i  +  4?/2  +  l/s) •     (Simpson's  first  rule.) 
o 

3.  Rule  for  parabola  of  the  third  order  with  four  ordinates. 
Area  =  ——  (i/i  +  3?/2  +  3?/3  +  2/4).     (Simpson's  second  rule.) 

o 

4.  Rule  for  parabola  of  the  fifth  order  with  five  ordinates. 
Area  =  -J^i'^Vi  +  32?/2  + 12?/3  +  32?/4  +  ly^)  ■ 

5.  Rule  for  parabola  of  the  fifth  order  with  six  ordinates. 

5  Ax 
Area  =  -—  {19yi  +  75?/2  +  50ys  +  50yi  +  Ibys  +  IQi/e) . 

6.  Rule  for  parabola  of  the  seventh  order  with  seven  ordinates . 
Area  =  ^  (41yi  +  216t/2  + 272/3 +  2721/4  +  27^6 +  2162/6  +  41^7)  ■ 

7.  Approximation  for  curve  with  six  ordinates. 
Area  =  -^^  {O'^yi  +  y-i  +  t/a  +  2/4  +  2/5  +  0*42/6) . 

8.  Weddle's  approximation  for  curve  with  seven  ordinates. 

3  Aa; 
Area  =  — -  {yi  +  52/2  +  2/3  +  67/4  +  2/5  +  62/6  +  2/7)  • 


24.  To  measure  any  curvilinear  area  by  means  of  the  tra- 
pezoidal rule. 

Rule. — To  the  sum  of  half  the  two  endmost  ordinates  add 
all  the  other  ordinates,  and  multiply  the  sum  by  the  common 
interval  ;  the  result  will  be  the  area.     Thus, 

\ydx=Ax(y'-y^  +  y2  +  ys  .  .  .  .  +  2/n-i) 

Remark. — In  Bhipbuilding  work  it  is  very  often  convenient 
to  perform  the  additions  in  the  above  rule  mechanically,  by 
measuring  off  the  ordinates  continuously  on  a  long  strip  of 
paper,  and  measuring  the  total  length  on  the  proper  scale. 
This  rule  is  only  approximate,  but  it  is  especially  useful  for 
getting  the  areas  of  the  transverse  sections  in  the  first  rougl? 
calculations  of  trim,  displacement,  etc. 

25.  To  measure  any  curvilinear  area  by  means  of  Simpson's 
first  rule. 

Rule. — To  the  sum  of  the  first  and  last  ordinates  add  four, 
times  the  intermediate  ordinates  and  twice  all  the  dividing 


44  MENSURATION   OF    CURVILINEAR   AREAS. 

oidinates  ;  multiply  the  final  sum  by  ^,  the  common  interval :  the 
result  will  be  the  area.     Thus 


/^ 


ydx  =  ~{y,  4  43/2  +  22/3  +  4^,  +  2^3 +  4^„_i  +  yn\ 


Remark. — The  number  of  intervals  in  this  rule  must  be 
even.  The  ordinates  which  separate  the  parabolas  into  which 
the  figure  is  conceived  to  be  divided,  are  called  dividing  ordi- 
nates, and  all  the  other  ordinates  except  the  two  endmost  ones 
are  called  intermediate  ordinates. 

26.  To  measure  any  curvilinear  area  by  means  of  Simpson' f 
second  rule. 

KULE. — To  the  sura  of  the  two  endmost  ordinates  add  three 
times  the  intermediate  ordinates  and  twice  all  the  dividing 
ordinates  ;  multiply  the  final  sum  by  |,  the  common  interval,  and 
the  result  will  be  the  area.     Thus 


/ 


ydx  =  -^-(//,  +  3?/2  +  3^3  +  2?/,  +  3y,  .  .  .  .  +  3y„_i  +  yn). 


The  number  of  intervals  in  this  case  must  be  a  multiple  of  three. 
He/nark. — The  sequence  of  the  multipliers  in  the  two  fore- 
going rules  is  obvious.     Thus  in  the  first  rule  the  simple  multi- 
pliers are  1.4.  1,  but  they  are  combined  thus : — 


1.4.1 

1.4.1 


1.4.1 


1.4.1 

1.4.1 


1.4.1 


1.4.2.4.2.4    4.2.4.2.4.1 

In  the  second  rule  the  multipliers  are  1.3.3.1. 

1.3.3.1 

1.3.3.1 

1.3.3.1  

1.3     3.1 


1.3.3.1 


1.3.3.2.3.3.2.3.3  3.3.2.3.3.1 

And  iii  the  same  way  the  multipliers  to  measure  any  curvi- 
linear area  may  be  obtained  from  the  table  on  p.  43. 

Simpson's    first    rule    is    superior   to   the   second   rule  in 
accuracy  as  well  as  simplicity. 

27.     To    measure   any    curvilinear   area    when    subdivided 
intervals  are  used. 

\st.   When  Si'mpson's  first  rule  is  used. 

Rule. — Diminish  the  multiplier  of  each  ordinate  belonging 
to  a  set  of  subdivided  intervals  in  the  same  proportion  in  which 


MENSURATION    OF    CURVILINEAR    AREAS. 


45 


the  intervals  are  subdivided.  Multiply  each  ordinate  by  its 
respective  multiplier  as  thus  found,  and  treat  the  sum  of  their 
products  as  if  they  were  whole  intervals  ;  that  is,  multiply  the 
sum  thus  found  by  i  of  a  whole  interval,  and  the  product  will 
be  the  area. 

27id.   When  Simpson's  second  rule  is  used. 

Rule. — Proceed  as  in  the  first  rule,  but  multiply  by  §  of  a 
whole  interval  for  the  area. 

Exaviple  to  Simpson's  First  Ride. — The  series  of  multipliers 
for  whole  intervals  being  1  .  4.2.4.2,  &c.,  those  for  half- 
intervals  will  be  f,  .  2  .  1  .  2  .  1,  &c.,  and  for  quarter-intervals 

Remark. — When  an  ordinate  stands  between  a  larger  and 
a  smaller  interval,  its  multiplier  will  be  the  sum  of  the  two 
multipliers  which  it  would  have  had  as  an  end  ordinate  for  each 
interval.  Thus  for  an  ordinate  between  a  whole  and  a  half 
interval  the  multiplier  is  ^  + 1 
quarter  interval  ^-  +  J  =  f . 


Table  of  Multipliers  when  Subdivided  Intervals 
ARE  used. 

Simpson's  First  Rule. 

Ordinates         |0  |l 

2 

2^ 

2i 

2 
3 

3  j3| 

3i 

4 

!4 

6 

H 

2 

7 
1~ 

7h 
2 

8 
1 

Multipliers 

1    4 

1^ 

n 

Ordinates 

0 

i 

1 

H 

2 

2^:3 

2~|U 

4 
4 

5 
1^ 

2 

6 
1 

61 
I 

H 
i 

1 

7 
1 

Multipliers 

12 

1 

2 

1 

Ordinates 

o|i. 

2 

2^:3 

3}3| 

3f 

4 

H 

H 

4| 

^ 

4.^ 

6 

Multipliers 

1  u 

u 

2      f 

li^ 

1 

1% 

1 

^ 

2 
3 

k 

1 

i 

Simpson's  Second  Rule.                                     1 

Ordinates 

0 

1 

2 
3" 

3 

1 

4    4i;5  15^6 
Ujl*:2    2i2^ 

6| 
2f 

8 

6S 
31 

6t 

3| 
f 

1 

7 

J 
4 

J 

6 

1 

Multipliers 

1 

3 

1 

Ordinates 

0 

Multipliers 

1 

1 

1 
J 

1 
1 

i! 
1 

J 

2 

Jjjijj 

2^  3    31  ^  4 

J 

if. 

1 

Ordiftates 

Multipliers 

u 

t  1 

1 

1 

Note. — The  ordinates  in  this  table  are  numbered  the  same 
as  if  they  were  the  number  of  intervals  from  the  origin. 


46 


MENSURATION     OF     AREAS    AND    PERIMETERS. 


ThomsoiVs  Rule  may  be  used  when  subdivided  intervals  are 
used  at  each  end  ;  the  advantage  being  that  all  multipliers  except 
the  three  end  ones  are  unity  ;  so  also  in  the  common  multiplier. 
Thus  the  ordicates  should  be  multiplied  by  ^,  J,  ^,  1,  1,  1,  ... 
...  ,  1,  1,  i^,  ^,  i ;  the  spacing  of  the  three  ordinates  at  each  end 
being  one  half  that  elsewhere. 

28.  To  calculate  the  area  separately  of  one  of  the  two 
divisions  of  a  parabolic  figure  of  the  second  order.  (Fig.  66.) 
Rule. — To  eight  times  the  middle  ordinate  add  five  times 
the  near  end  ordinate,  and  subtract  the  far  end  ordinate  ; 
multiply  the  remainder  by  3^2  *^e  common  interval  :  the  result 
will  be  the  area. 

Note. — The  near  end  ordinate  is  the  ordinate  at  the  end  of 
the  division  of  which  the  area  is  to  found. 

Ex. :   In   figure   abcd    let  it  be  required 

to  find  the  area  of  the  division  acef.     Let 

y^  =  the  near  end  ordinate,  ^2  =  ^^^  middle 

\y/         ^  S^s  ordinate,   and   ^3  =  the   far   end   ordinate  ; 

\  Asc  A    tx     I     then  \ydx  =  -j^{oyi  +  8^2  -  y&). 


Fig.  66. 


y 


12 


29.  To  measure  an  area  bounded  by  an  arc  of  a  '^plane  curve 
and  two  radii.     (Fig.  67.) 

Fig.  67.  RuLE. — Divide  the  angle  subtended  by 

the  arc  into  any  number  of  equal  angular 

intervals  by  means  of  radii.    Measure  these 

radii     and     compute     their     half-squares. 

Treat  those  half -squares  as  if  they  were 

ordinates  of  a  curve  by  Simpson's  first  or 

second   rule,   as   the   number   of  intervals; 

C      may  require. 

Note. — The   common   interval   must   be   taken   in   circular 

measure.     (See  pp.  8  and  9.) 

Ex. :    In   the  figure  ABC  let  vi,   r^,   ra,   Vi,  rs  =  the  radii, 

-dd  =:  the  area ;  then 

(ri2  +  4^22  ^  2rs^  +  W  +  r.5-)  A9 


n 


Id 


30.  To  measure  any  curvilinear  area  by  nteans  of  Tcheby- 
cheff's  rule. 

Rule. — Find  the  middle  point  of  base,  and  from  it  set  off, 
along  the  base,  and  in  both  directions,  distances  equal  to-  the 
half  length,  of  base  multiplied  by  the  constants  given  in  the 
Schedule  below.  Erect  ordinates  at  the  points  so  obtained 
and  measure  them.  Their  sum,  divided  by  the  number  of 
ordinates,  and  multiplied  by  the  length  of  base  is  the  area 
required. 


MENSURATION    OF    AREAS    AND    PERIMETERS. 


47 


Schedule. 

Number  of 
Ordinates  used. 

Distance  of  Orainates  from  Middle  o!  Base  In 
Fractions  of  Half  the  Base  Length. 

2 
3 
4 
5 
6 
7 
9 

•5773 
0,  ^7071 

•1876,  -7947 

0,  -3745,   -8325 

•2666,  ^4225,  -8662 

0,  -3239,  ^5297,  -8839 

0,   -1679,  -5288,  -6010.  -9116 

Note. — As  evident  from  the  Schedule,  there  is  an  ordinate 
at  the  middle  of  base,  only  when  an  odd  number  of  ordinates 
is  employed. 


Examples. — With  four  ordinates. 

Let  ABCD  be  the  figure.  Bisect 
the  base  AB  at  E.  Calling  the  half 
length  of  base  b,  set  off  EF  and  ef' 
equal  to  ^1876  h  and  eg  and  eg'  equal 
to  ^7947  h.  Erect  ordinates  GL,  fk, 
f'k',  g'l'  at  G,  F,  f',  g'  ;  and  call 
them  7/1,  3/2,  ys,  and  y*. 


(Fig.  68.) 


K if *+« {f- M 


Then  area  of  figure  abcd 


Vi  +  y-2  +  ys  +  Vi 


X  26. 


With  five  ordinates. 
Fig.  G'J. 
E'       K 


(Fig.  69.) 


-6 -H" 


■b" 


As  before,  let  ABCD  be  the 
:  figure,  e  the  middle  of  base,  and  b 
its  half-length.  Set  off  EF  and  ef' 
equal  to  -3745  b  and  EG  and  eg' 
equal  to  -8325  6,  and  erect  ordinates 
B  at  G,  F,  e,  f',  and  g',  calling  them 
2/1,  2/2,  2/3.  2/4,  2/«. 


Then  area  of  figure  ABCD  =  ^^  +  ^^^  +  V^  +  V^  +  V^  ^  26. 

5 

Note. — This  rule  can  be  used  for  calculating  displacements, 
and  fewer  ordinates  are  required  for  the  same  degree  of 
accuracy  than  if  Simpson's  rule  is  used.  Ten  ordinates  are 
usually  employed  instead  of  twenty-one,  the  rule  for  five 
ordinates  being  applied  separately  to  each  half  of  the  ship.  It 
is  also  of  great  assistance  in  preparing  cross  curves  of  stability. 
If  eight  ordinates  are  used  (four  repeated),  the  following 
"  Simpson "  sections,  assumed  numbered  from  1  to  21,  can 
be  utilized  :    2,  5,  7,  10,  12,   15,  17,  20. 


iS 


MENSURATION    OF    AREAS    AND    PERIMETERS. 


31.  To  measure  any  curvilinear  area  by  three  ordinate^ 
irregularly  spaced. 

Rule. — Let  abcdef  (Fig.  70)  be  the  curvilinear  area,  whose 
ordinates  ab,  fc,  ed,  are  yi,  2/2,  and  ^3.  Let  af  =  7t,  and  YE  =  hh, 
where  fe  is  a  ratio. 

^^^[7^17.(2-7.)  +  y,  [h  +  lY  +  y^[2U-l)] 

Note.—li  AF  =  2fe,  so  that  7c  =  I,  ' 

Area    =    -^  (7/1  +  87/2)         \ 


32.  To  find  the  area  hetxoeen   the  first   tivo   ordinates  of 
a  curvilinear  area  given  three  ordinates  irregularly  spaced. 

EuLE. — The  area  included  between  the  ordinates  ar  and  cf. 

"  6A:(/f+i)  ^y'  ^'  ^^^'  +  ^^  +  '^'^  ^^^  +  ^^  ^^'  +  ^^ "  y^^ 

Note. — If  AF  =  2fe,  so  that  h  =  l. 


Fig.  70. 


AF 


Area  =  —  (7  7/i  +  15?/2-4?/3) 
If  AF  =  |fe,  so  that  h  =  2, 

Area  =  -11(167/1  +  217/2-7/3) 


33.  To  obtain  a  general  exjJression  for  the  length  of  any 
plane  curve. 

Using  cartesian  co-ordinates  the  length  intercepted  between 
two  points  whose  '  x  '  co-ordinates  are  a  and  h  is  equal  to  the 

definite  integral  /      /y   1  +  (y  )    .  dx.    This  may  be  obtained 

by  Simpson's  rules  in  a  similar  way  to  the  area  ;    the  '  ordinate  ' 

in  this  case  being  replaced  by  the  value  of  ^  1  +  (-^)    or 

sec.  <p,  and  the  common  interval  being  measured  along  ox. 


34.  To  find  approximately  the  length  of  any  plane  curve, 
(Fig.  71.) 

If  the  curve  is  rather  irregular,  divide  it  by  the  eye  into 
any  number  of  fairly  flat  aarcs  ;  join  the  extremities  of  each 
of  these  arcs  by  choi-ds.  The  sum  of  the  leng-fch  of  each  of 
these  arcs  found  by  the  following  rule  will  be  the  total  length 
of  the  curved  line. 


MENSURATION    OF   SOLIDS.  4^ 

RuLK. — Draw  a  tangent  to  the  curve  at  each  of  its  ex- 
tremities ;  then  take  the  sum  of  the  two  distances  from  the  point 
of  intersection  of  the  two  tangents  to  the  extremities  of  the 
curve,  together  with  twice  the  length  of  the  chord  ;  divide  the 
result  by  3  for  the  lenja^th  of  the  arc. 

Ex.  (fig.  71):  Let  acb  be  one  of 
the  arcs,  and  ab  a  chord  joining 
the  two  extremities,  and  at,  bt" 
tangents  to  the  curve  at  its  extremi- 
ties, cutting  each  other  in  D  ;  then 
the  length  of  the  curve 

ACB  =  i  (ad  +  DB  +  2ab). 

Alternatively,  for  a  flat  curve  see  Rule  III,  p.  38. 

35.  To  fi)id  the  perimeter  of  an  ellipse  of  moderate 
eccentricity. 

Rule. — If  2a  is  the  major  and  26  the  minor  axis,  where  -  is 

not  very  small,  then  the  perimeter  is  equal  to  —  (3a^  +  6')  very 

nearly. 

36.  To  find  the  length  of  the  evolufe  of  a  curve. 

Rule. — The  length  of  the  arc  of  the  evolute  is  equal  to  the 
difference  between  the  lengths  of  the  tangents  drawn  from 
either  extremity  of  the  arc  to  the  involute. 


II.   Mensuration  of  Solids. 

1.  To  find  the  volume  of  any  parallelopiped,  pristn,  or 
cylinder.     (Fig.  72.) 

Rule. — Multiply  the  area  of  the  base  by  the  porpendicular 
height  ;  the  result  will  be  the  volume. 


Pig.  72. 


2.  To   find   the  vohime   and   slant    surface   of   a   cone   or 
pyramid.      (Fig.  73.) 

Fig.  73. 


Rule. — Afultiply  the  area  of  the  base  by  ^  the  perpen- 
dicular height  ;    tlse  product  will  be  the  volume.     The  slant 

E 


60  MENSURATION    OF   SOLIDS. 

surface    is    equal    to    the    periineiter    of    the    base   multiplied 
by  half  the  slant  height. 

3.  To  find  the  volume  and  slant  surface  of  the  frustum 
of  a  cone  or  pyramid.     (Fig.  74.) 

EuLE. — To  the  sum  of  the  areas  of  the  two  ends  add  the 
square  root  of  their  product  ;  this  final  sum  being  multi-t 
plied  by  ^  of  the  perpendicular  height  will  give  the  volume. 
The  slant  surface  is  the  product  of  the  sum  of  the  perimeter 
of  the  two  ends  and  half  the  slant  height. 

Fig.  74. 


4.  To    find    tJie    volume    of    a    wedge  whose    base    is    a 
rram.      (Fig.   75.) 
Fig.  75.  Rule. — Add.  the    length    of    the 

. J  edge  to  twice  the  length  of  the  base ; 

/|  \\     P^ U    niultiply  the  sum  by  the  width  of 

/"'"  V     i-''  '"■'■  ^    the  base  and  the  product  by  ^  of  the 

perpendicular  height  :   the  result  will  be  the  volume. 

5.  To  find  the  volume  of  a  prismoid.     (Fig.  76.) 

EuLE. — To  the  sum  of  the  areas  of  the  two  ends 
Fig.  76.    ^dd  four  times  the  area  of  a  section  parallel  to  the 
base  and  equally  distant  from  both  ends  ;   the  sum 
being  multiplied  by  ^  the  perpendicular  height  will 
give  the  volume. 

6-    To  find  the  volume  and  stirface  of  a  sphere  or  globe. 
(Fig.  77.) 
Pj^  ^^  Rule. — Multiply    the   cube   of   the   diameter   by 

-^  or  •52o6  ;    the   product   will   be   the   volume.      To 

obtain  the  surface,  multiply  the  square  of  the  diameter 
by  TT  or  3-1416. 

7.  To  find  the  volume  and  surface  of  the  segment  of  a  sphere. 
(Fig.  78.) 

Rule. — Add  the  square  of  the  height  to  3  times 
the   square   of   the   radius   of   the   base  ;    that   sum 
Fig.  78.  ir 

multiplied  by  the  height  and  that  product  by   ^  or 

•5236  will  give  the  volume.  To  obtain  the  surface, 
multiply  the  diameter  of  the  whole  sphere  by  the 
heit^ht  of  the  segment  and  that  product  by  w  or 
3-1416. 


MENSURATION    OF   SOLIDS. 


51 


8.  To  find  the  volume  and  surface  of  a  zone  of  a  sphere. 
(Fig.  79.) 

Rule, — To  the  sum  of  the  squares  of  the  radii  of 
the  two  ends  add  ^  the  square  of  tlio  height  ; 
multiply  the  sum  by  the  height  and  that  result  by     ^^'^ 

-   or  l*o708  :    the  result  will  be  the   volume.     To  fiiiiHilHiiaii 

obtain   the   surface,   multiply   the   diameter   of  the  ""^ — ^ 
whole  sphere  by  the  height  of  the  zone,  and  that 
product  by  v  or  3*1416. 

9.  To  find  the  volume  and  surface  of  a  cylindrical  ring. 

Rule. — To  the  thickness  of  the  ring  add  the  inner  dia- 
meter ;  multiply  that  sum  by  the  square  of  the  thickness,  and 

the  product  by  —  or  24674  :   the  result  will  be  the  volume. 

To  obtain  the  surface,  multiply  the  sura  of  the  inner  diameter 
and  thickness  by  the  thickness,  and  that  product  by  it'^  or  9-87. 


Table  to  find  the  Volume  and  Surface  of  any 
Regular  Polyhedron. 


volume.  A  =  area.  L  =  linear  ec 

r  =  radius  of  inscribed  circle. 


No.  of 
Sides 


12 

20 


No.  of 
Edges  in 
each  side  I 


Name 


Tetrahedron 
Hexahedron ' 
Octahedron 
Dodecahedron 
Icosahedron 


1-732051 
6-000000 
3-464102 
20-645729 
8-660254 


V  =  l3x 


•117851 
1-000000 

•471405 
7-663119 
2-181695 


r=iiX 


•204124 
•600000 
•408248 
1-113516 
•755760 


Or  cube. 


10.  To  find  the  volume  of  an  ellipsoid.     (Fig.  80.) 
Rule. — Multiply    the    product     of    the    three       fio.  80. 

principal  axes  by  -  or  "5236  :    the  result  will  be 

the    volume. 


11.  To  find  the  volume  of  the  segment  of  an  ellipsoid  of 
revolution  when   the   base   is   circidar.      (Fig.    81.) 

Rule. — Take    double    the    height    of    tlie    segment    from 


MENSURATION    OF    SOLIDS. 

three  times  the  length  of  the  fixed  axis  ;    multiply 
the    difference    by    the    square   of    the   height,   and 

that    product    by    ^   or    '5236  :     then    that    result 

6 
multiplied  by  the  square  of  the  revolving  axis  and 
the  product  divided  by  the  square  of  the  fixed  axis 
will  give  the  volume. 


12.  To  -find  the  volume  of  the  segment  of  an  ellipsoid  of 
revolution  when  the  base  is  elliptical.     (Fig.   82.) 

Hule. — Take  double  the  height  of  the  segment  from  three 

times  the  length  of  the  revolving  axis  ;  multiply 

Fig.  82.       the  difference  by  the  square  of  the  height,  and 

that  product  by    —  or  '5236  :    then  that  result 

multiplied  by  the  fixed  axis,  and  the  product 
divided  by  the  revolving  axis,  will  give  the 
volume. 


13.  To    find    the   volume    of    the   middle   frustum    of   an 
ellipsoid  of  revolution  when  the  ends  are  circular.     (-Fig.  83.) 

Fig.  83.  RuLE. — Multiply  the  sum  of  the  square  of  the 

'     '       middle  diameter  and  one-half  the  square  of  the 

f  I  '^^m\  diameter  of  one  end  by  the  length  of  the  frustum, 

^-\^^^y   and  that  product  by  -  or  "5236  for  the  volume. 

b 


14:.    To   find   the   volume   of   the   middle   frustum   of   an 
ellipsoid  of  revolution  when  the  ends  are  elliptical.  (Fig.  84.) 

IluLE. — To  twice  the  product  of  the  transverse  and  con- 
jugate diameters  of  the  middle  section,  add  the 
Fig.  84.        product  of  the  transverse  and  conjugate  diameters 
of  one  end  ;  multiply  the  sum  by  the  height  of 

the  frustum,  and  that  product  by  —  or  •2618  : 

the  result  will  be  the  volume. 


15.  To  find  the  volume  of  a  paraboloid.   (Fig.  85.) 

Fig.  85.  RuLE. — Multiply  the  square  of  the  diameter 

of  the  base  by  the  perpendicular  height,  and  the 

result  by    -  or  '3927  ;    the  product  will   be  the 
8 


MENSURATION    OF    SOLIDS.  53 

16.  To  find   the  volume  of   the  frustum   of  a  paraboloid 
when  its  ends  are  perpendicular  to  its  axis.     (Fig.  86.) 

Rule. — Multiply  the  sum  of  the  squares  of     Fia.  86. 
the  diameters  of  the  two  ends  by  the  height  of 

the  frustum  ;     the  product  multiplied  by    -    or 

8 
•3927  will  be  the  volume.  "flUilUl 


/ 


17.  To  find  the  volume  of  any  solid  of  revolution. 

(I)  The     volume    is    represented    by  the    definite    integral 
h 

■tr  if- .  dx,  where  Ox  is  the  axis  of  symmetry. 

0 

Rule. — Divide  the  length  of  the  axis  into  a  convenient 
number  of  equal  parts.  Measure  the  ordinates,  and  treat 
their  squares  as  if  they  were  the  ordinates  of  a  plane  curvq 
of  the  same  length  aa  the  solid  ;  the  area  of  this  curve 
multiplied  by  it  or  3'1416  will  be  the  volume  required. 

(II)  If  the  position  of  the  centre  of  gravity  of  the 
generating  area  is  known,  the  following  method  is  applicable. 

Rule. — Multiply  the  area  of  the  generating  section  by  the 
distance  of  its  centre  of  gravity  from  the  axis  ;  2Tr  or  6283 
times  the  product  will  be  the  volume  required. 

Example. — Find  the  volume  of  the  solid  generated  by  the 
revolution  of  an  equilateral  triangle  about  its  base. 

If  2a  be  the  side  of  the  triangle,  its  area  is  V3  .  a^ ;  and  its 
a 
centre   of  gravity  is  —~  from   the   base.      Hence   the   volume 
V  3 
—  a 

required  is  2ir  X   Vda^  x    -=  =  27ra'. 

18.  To  Pleasure  the  volume  of  any  solid, 
(I)    To  measure  the  volume  in  slices. 

Rule. — Take  one  of  tho  plane  surfaces  as  the  base,  and 
divide  the  mass  into  slices  parallel  to  that  base  and  suflBciently 
thin  as  to  be  able  either  to  neglect  or  account  separately  for 
the  curvature. 

Then  take  tho  volumo  of  each  slice  separately,  and  add 
them  together  for  the  whole  volume,  taking  account  of  the 
curvature  in  this  addition  if  necessary. 


54 


MENSURATION   OF   SOLIDS. 


Fig.  87. 


(II)  To  measure  the  volume  by  the  rules  applicable  to  the 
area  of  a  plane  curve.     (Fig.   87.) 

EuLE. — ^Take  a  straight  line  in 
tho  figure  as  a  base  line,  or  line  of 
abscissa,  and  divide  the  figure  along 
that  line  into  any  number  of  equal 
parts,  and  measure  tho  areas  of  the 
plane  sections  at  those  points  of  divisdon  by  the  rules 
applicable  to  the  area  of  a  plane  curve. 

Then  treat  the  areas  thus  found  as  if  they  were  the 
ordinates  of  a  plane  curve  of  tho  same  length  as  the  figure, 
and  the  area  of  this  will  be  the  volume  of  tho  solid. 


Exanvple.    (See  fig.  87.) 


Number  of 
Sections 

Areas  of  Sections 
in  square  feet 

Multipliers 

Products 

I 

5 

1 

5 

2 

10 

4 

40 

3 

15 

2 

30 

4 

20 

4 

80 

5 

25 

1 

25 

Ax 


180 
=     2 


Area  =  360  cubic  feet. 


RemarJe. — The  volume  is  above  assumed  to  be  represented  by 


J    0 


the  definite  integral  /      K  .  dx,  where  A  is  the  area  of  any  section 

J    0 

perpendicular  to  the  base  line  ox.  The  volume  may  also  be 
represented  by  the  double  integral  \\2,dx.dy  taken  over  the 
area  of  the  base,  the  a  axis  being  supposed  perpendicular  to 
the  base. 


(Ill)  To  measure  the  volume  by  JDr.  Woollcy^s  method, 
(Fig.  88.) 

EuLE. — ^Take  a  straight  line  in  the  figure  as  a  base  line, 
and  divide  the  figure  along  that  line  by  an  odd  number  of 
parallel  and  equidistant  planes  perpendicular  to  the  base. 
Then  divide  the  figure  horizontally  in  the  same  way  by  a 
number  of  plane  sections  parallel  to  the  base.  Then  take 
ordinates  Jat  the  intersections  of  the  horizontal  with  the  vertical 
plane  sections  in  their  consecutive  order,  and  treat  them  as 
follows  :— 


MENSURATION    OF    SOLIDS. 


55 


(1)  Neglect  absolutely  all  ordinatea  wiiich  are  odd  in  both 
planes  of  section. 

(2)  Neglecting  the  outside  rows  of  ordinates,  double  every 
ordinate  which  is  even  in  either  or  both  planes  of  section,  and 
add  them  together. 

(3)  Add  to  this  the  simple  sum  of  all  the  even  ordinates  in 
the  outside  rows. 

(4)  Multiply  this  final  sum  by  |  of  the  product  of  the 
common    vertical    interval,    by    the  Pxo,  88. 
common  horizontal  interval,  and  the  p;— — j's- — p^- — j^;- — ^ 
result  will  be  the  volume.                  [\T"T^"""^^^""iwT"io 

Ex.:  In  the  accompanying^ '         '  "^     '  '^'    ''^^'    ' '"'  ' 

the  multiplier  for  each  or( 

shown  above  it,  so  that 

sum  of  the  products  of  the  ordinatos 

by  their  respective  multipliers,  v  ==        "        '      .  " 

the    volume,    and  Av'  =  the    common    vertical    interval,  and 

Ax  =  the   common  horizontal  interval,  then 

2  {s  X  Ax'  X  Aa:) 
-'  =  - 3 ■ 

Remarh. — This  method  is  inferior  in  accuracy  to  that 
obtained  by  a  double  application  of  Simpson's  rules. 


19.  To  measure  the  volume  of  a  wedge-shaped  solid 
bounded  on  one  side  by  a  curved  surface.     (Fig.  89.) 

The  volume  is  represented  by  the  double  integral  jji^** 
dx  .  do,  where  r  is  a  radius  from  the  edge.  0  is  the  angle  between 
the  radius  and  the  plane  of  the  base,  and  ox  is  parallel  to  the  edge. 

EuLE. — Divide  the  figure  longitudinally  by  a  number  of 
planes  radiating  from  the  edge  at  equal  angular  intervals,  and 
also  divide  the  length  of  figure  into  yig  69. 

a    number    of    equal    intervals    for  \^ 

ordinates,    and    treat    each    of    thc^^ -^ "^      " 

radiating   planes   as    follows  :  — 

(I)  Measure  the  ordinates  as  if 
for  taking  the  areas  of  the  several 
planes,  but  instead  of  the  ordinates 
themselves  compute  their  half- 
squares,  and  treat  them  as  if  they 

were  the  ordinates  of  a  plane  curve  of  the  same  length  as 
the  figure.  The  result  of  this  calculation  is  called  the  moment 
of  the  radiating  plane. 

(II)  Treat  the  moments  of  the  radiating  planes  as  if  they 
were  the  ordinates  of  a  curve,  but  taking  the  common  angular 
interval  in  circular  measure. 


56 


MENSURATION    OF   SOLIDS. 


Example.    (See  fig.  89.) 


No.  of  Planes 

Moments  of  the 
Eadiating  Planes 

Multipliers 

Products 

1 
2 
3 
4 
5 

105 
110 
115 
120 
125 

1 
4 
2 

4 

1 

105 
440 
230 
480 
125 

angular  interval 
3 


1380 
=  •0291 

1380 
12420 
2760 


Volume  =  40-1580 

20.  To  find  the  mean  sectional  area  of  a  solid. 

Rule. — Divide  the  volume  of  the  solid  by  its  length  ;  the 
result  will  b©  the  mean  sectional  area. 

21.  To  set  off  the  correct  form  of  a  mean  cross-section. 
Rule. — ^Divide  the  figure  longitudinally  by  a  number  of 

horizontal  planes  ;  take  the  mean  breadth  of  each  of  the 
horizontal  planes  and  set  them  off  perpendicular  to  a  fixed 
straight  line,  and  at  the  sam(3  height  as  their  corresponding' 
planes  in  the  solid  :  a  line  passing  through  the  ends  of  these 
mean  breadths  will  be  the  correct  form  of  the  mean  sectional 
area  of  the  solid. 

Note. — ^The  mean  breadth  of  a  plane  curve  is  found  by 
dividing  the  area  of  the  curve  by  its  length. 


22.    To  find  the  volume  of  a 

fourway  'piece  of  piping. 

Let  r  (fig.  90)  be  the  radius 
of  the  piping  and  I  and  V  the 

lengths. 


/ 


Then  volume  =  Trr^  [l +1'  -  §r). 


23.  To  find  the  surface  of  any  solid  of  revolution. 
(I)  The    surface    is    represented    by    the    definite    integral 
h 
tty  .  ds,  where  ds  is  an  element  of  arc  of  the  generating  curve. 

0 


MENSURATION   OF   SOI  JDS.  57 

Rule. — Divide  the  perimeter  of  the  generating  curve  into 
a  convenient  number  of  equal  arcs.  Measure  the  ordinatoa 
at  the  points  of  division,  and  treat  them  as  if  they  were  the 
equidistant  ordinates  of  a  curve,  with  the  common  interval 
equal  to  the  length  of  the  arcs.  The  area  of  tliis  curvo 
multiplied  by  2ir  or  6"283  will  be  the  area  of  the  surface. 

(II)  If  the  position  of  the  centre  of  gravity  of  the  peri- 
phery of  the  generating  curve  is  known,  the  following  method 
is  applicable  : — 

Rule. — Multiply  the  length  of  the  arc  of  the  generating 
curve  by  the  distance  of  the  centre  of  gravity  of  the  arc 
from  the  axis  ;  lir  or  6283  times  the  product  will  be  the  area 
of  the  surface. 

Example. — Find  the  surface  of  the  solid  generated  by  the 
revolution  of  an  equilateral  triangle  about  its  base. 

If  2a  be  the  side  of  the  triangle,  its  perimeter,  exclusive  ofthe 
axis,  is  4<2  ;  and  the  centre  of  gravity  of  the  two  sides  is  V3fl/2 
from  the  base. 

Hence  the  surface  required  is  27r  x  4a  x   v'S  a\1  =  4  VS  ire?. 


24.  To  find  the  area  of  any  surface. 

(I)  Exact  metliod. — The  area  is  given  by  the  double  integral 
J  J  sec  S  .  da;  .  dy^  where  Q  is  the  angle  made  by  the  surface  with 
the  xy  plane.  This  is  equal  to  J  J  »/'{l  +  tan'^  <p  +  tan'^  «//)  . 
dx  .  dy,  where  ^  and  ^  are  the  angles  made  with  the  z  axis  by 
the  sections  of  the  surface  with  the  xz  and  yz  planes. 

Rule. — Take  a  straight  line  in  the  figure  as  base  line  and 
divide  the  figure  along  that  line  by  a  convenient  number  of 
parallel  and  equidistant  planes  perpendicular  to  the  base  ; 
call  these  the  vertical  sections.  Then  divide  the  figiiro 
horizontally  in  the  same  way  by  a  number  of  plane  section^ 
parallel  to  the  base.  At  the  intersections  of  the  two  sets  of 
sections  measure  tan  (p  and  tan  xj/f  <p  and  t|/  being  the  angles  made 
in  the  sections  by  the  tangents  to  the  curves  with  the  base. 
Evaluate  Vl  +  tan*  <t>  +  tan'^ «//  at  each  intersection.  Treat  this 
as  the  ordinate  of  a  solid,  and  proceed  to  find  its  volume  by  any 
of  the  rules  given  above.  The  result  is  the  area  required.  It  is 
desirable  that  no  part  of  the  surface  should  be  approximately 
perpendicular  to  the  base. 

Example. — A  portion  of  a  ship's  side  is  bounded  by  two 
sections  40  feet  apart,  and  two  waterlines  8  feet  apart.  The 
tangents  of  the  angles  (0)  made  with  the  middle  line  at  the 
flections  and  at  one  situated  midway  between  them,  and  the* 


58 


MEXPURATI-ON   OF   SOLIDS. 


tangents  of   tlie  angles    (f)   made   with   the   middlo  lino  at 
the  two  watorliuGS,  and  At  one  midway,  are  as  follows:  — 


W.L. 

1 
2 
3 

Section  1 

Section  2 

Section  3 

tan'/>     1    tan'f' 

tan<P 

tan'/' 

tan'/> 

tau'/' 

•30      j      -21 
•21      i      ^22 
•]0      1      -22 

•35 
•31 
•14 

•16 
•17 

•18 

•48 
•44 
•25 

•06 
•07 
•09 

nd  v^l  +  tan^  <p  +  tan^  4'  in  each  case,  and  proceed  as  in  the  table  below  :- 


II 

?»2 

Section  1 

13 

2 

Cm 

Section  2 

1 

Section  8 

s 

Vl+tan2  0+tan2'/' 

V  1+^2"^ + tan2> 

"H 

Vl  +  tan«(^  +  tan2i|/ 

£ 

1 

1-06 

1-06 

107 

1-07 

111 

1-1 

4 

1^04 

4-16 

1-06 

4-24 

1^10 

4.4 

1 

1^03 

1-03 

7-25 

1 

1-03 

1-03 

6-34 
4 

1-02 

1-0 
6-5 

7-25 


25-36 


6-5i 
=  39^] 


20 


The  area  of  the  curved  surface  is  39-14  x  -jt  x  0  =  348  square 
feet.  ^       ^ 


(II)    Approximate  method. 

Rule. — Take  girths  along  (say)  the  vertical  sections  at 
equidistant  intervals.  For  each  section  in  the  half -breadth 
plan,  note  the  angle  at  wjliich  the  various  waterlines  cro^s, 
and  estimate  the  mean  slope  of  the  waterlines  surroundingi 
the  surface  under  question.  The  secant  of  this  mean  anglo 
with  the  middle  lino  is  termed  the  modifying  factor,  and  ia 
multiplied  by  the  girth  concerned.  These  modified  girths 
are  then  regarded  as  the  ordinates  of  a  curve,  whose  area  ia 
the  surface  required. 

Example. — In  the  previous  example,  the  girths  at  Section? 
1,  2,  and  3  are  8-2,  83,  -uv.d  8*7  feet  respectively.  The 
respective  mean  angles  of  the  waterlines  with  the  middle 
line  are  '22,  '17,  and  '07.    Find  the  surface. 

The  first  modifying  factor  is  /sjl  +  ('22)2  or  1*02  ; 
similarly  the  others  are  1"015  and  1*0  approximately. 


CBNTHES  AND  MOMENTS  OF  FIGURES. 


69 


No. 

Girth 

Modifying 
Factor 

Modified 
Girtli 

Simpson's 
Multiplier 

Product 

1 
2 
3 

8-2 
83 
8-7 

102 

1015 

10 

8-4 

8-45 

8-7 

1 
4 

1 

8-4 

33-8 

8-7 

60-9 

20 
Area  of  surface  =  50-9  x  "«-  =  340  square  feet  approximately. 


CENTRES    AND    MOMENTS    OF    FIGUEES. 

To  FIND  THE  CeNTBES  OF  GRAVITY  OF  A  FEW  SPECIAL  FIGURES. 

1.  Triangle.     (Fig.  91.) 

Rule. — From  the  middle  points  of  any 
two  Bides  draw  lines  to  the  opposite  angle ; 
the  point  of  intersection  d  of  these  lines  is 
the  required  centre. 

Or,  trisect  the  line  joining"  the  middle 
point  of  one  side  with  the  opposite  vertex  ;    the  point  of 
trisection  nearer  to  the  base  is  the  required  centre. 


Fig.  92. 


2.  Trapezoid.     (Fig.  92.) 

Rule. — Bisect  ab  in  e  and  CD  in 
P  and  join  ef.  Produce  ab  beyond 
B  to  H,  making  Bii  =  CD,  and  pro- 
duce CD  beyond  c  to  i,  making  ci 
=  AB  ;  then  join  nr,  and  where  this  line  intersects  ef  is  the 
centre  of  gravity  G. 

Note. — EG  is  to  OF  as  2cD  -f  ab  is  to  2ab  -f  cd.     If  the 
angles  at  a  and  c  are  right  aisles,  the  distance  of  a  from  ac 

,    .      AB^  +  AB  .  CD  +  CD* 
''   ^'1'^"^   ^  3(AB  +  0D) 


3.   Quadrilateral.     (Fig.  93.) 

RuLC. — Draw  the  diagonals 
AD  and  cb  intersecting  in  E  ; 
along  CB  set  off  of  equal  to  eb, 
and  join  fa  and  fd  ;  the  centre 
of  the  triangle  afd  will  be  the 
centre  of  the  quadrilateral. 


Fia.  98. 


60 


CICNTIIES   AND  MOMENTS  OF   FIGURES. 


Fig.  93a. 


Fig.  94. 


Or,  bisect  the  diagonal  bd  (fig.  93a) 
at  E  ;  join  ea,  eg.  Make  ef  =  ^ea  and 
EG  =Jec.  Join  FG,  cutting  bd  at  h. 
Make  kg  =  hf.  Then  k  is  the  required 
centre. 


4.  Circular  arc.     (Fig.  94.) 

Rule. — Let  adb  be  the  circular  arc 
and  c  the  centre  of  the  circle  of  wliich 
it  is  a  part  :  bisect  the  arc  ab  in  d,  and 
Join  DC  and  ab  ;  multiply  the  radius  CD 
by  the  chord  ab,  and  divide  by  the 
length  of  tlie  arc  adb  ;  lay  off  the 
quotient  ce  upon  CD  ;  then  b  is  the 
centre  required. 


Fig.  95.  5.  Very  flat  curved  lino 

(approximato).  (Fig.  95.) 

KuLE.— Let  ADB  be  the 
arc  ;  draw  the  chord  ab, 
and  bisect  it  in  C;  draw  cd 

perpendicular  to  ab  ;  make  CE  equal  to  §  Df  CD  ;  then  E  will 

be  the  centre  required. 


6.  Sector  of  a  niide.     (Fig.  96.) 

Rule. — Let  abo  be  the  sector,  e  its 
centre;  multiply  the  chord  AB  by  §  of  the 
^  radius  CA ;  divide  the  product  by  the  length 
of  the  arc:  the  quotient  equals  the  distance 
CE  set  along  the  line  CD,  D  being  at  the 
bisection  of  the  arc  ab. 


7.  Sector  of  a  plane  circular  ring.     (Fig.  97.) 

Fig.  97.  RuLE. — Let  CA   be   the   outer  and  CE 

the  inner  radius  of  the  ring ;  divide  twice 
the  difference  of  the  cubes  of  the  inner 
^'  and  outer  radii  by  three  times  the 
difference  of  their  squares;  the  quotient 
will  be  an  intermediate  radius  CF,  with 
which  describe  the  are  ff,  subtending  the 
same  angle  with  the  sector  :  the  centre  H  of  the  circular  arc 
FF,  found  by  Rule  4,  will  be  the  centre  required. 


CENTRES  AND  MOMENTS  OF  FIGURES. 


8.  Circular  segment,     (Fig.  98.) 

Rule. — Let  c  be  the  centre  of  the 
circle  of  which  it  is  a  part  ;  bisect  the  arc 
AB  in  D,  and  join  CD  ;  divide  the  cube  of  ^^ 
half  the  chord  ab  by  three  times  the  area 
of  half  the  segment  adij  ;  set  ofiE  the 
quotient  ce  along  CD,  and  E  will  be  the 
centre  required. 


Fig.  98. 


61 


9.  Parabolic  half-segment.     (Fig.  99.) 

Rule. — Let  abd  be  a  half-segment  of 
a  parabola,  bd  being  part  of  a  diameter 
parallel  to  the  axis  and  ad  an  ordinate  con- 
jugate to  that  diameter — that  is,  parallel 
to  a  tangent  at  b.  Make  be  equal  to  f  bd, 
and  draw  ef  parallel  to  ad  and  equal  to 
I  aL'.  Then  f  will  be  the  centre  of  the 
half -segment. 


10.  Height  of  centre  'if  semicircle  or  semi-ellipse  from 
its   base. 

Rule. — Multiply  the  radius  of  the  semicircle  (or  that 
acmi-axis  of  the  ellipse  which  is  perpendicular  to  the  base) 
by  4,  and  divide  the  product  by  on. 

11.  Height  of  centre  of  parabola  from  its  base. 

Rule. — Multiply  its  vertical  height  by  2,  and  divide  the 
product  by   5. 

12.  Prism  or  cylinder  with  plane  parallel  ends. 

Rule. — Find  the  centres  of  the  ends  ;  a  straight  line 
joining  them  will  be  the  axis  of  the  prism  or  cylinder,  and  the 
middle  point  of  that  line  W(ill  be  the  centre  required. 

13.  Cone  or  pyramid. 

Rule. — Find  the  centre  of  the  base,  from  which  draw  a  line 
to  the  summit  ;  this  will  be  the  axis  of  the  cone  or  pyramid, 
and  the  point  at  J  fromi  the  base  along  that  line  will  be  tlio 
centre. 


14.  Hemisphere  or  hemi- ellipsoid. 

Rule. — The  distance  of  the  centre  from  the  circular  or 
elliptic  base  is  |  of  the  radius  of  the  sphere,  or  of  that  semi- 
axis  of  the  ellipsoid  which  is  parpendicular  to  the  base. 


62  CENTRES   AND  MOMENTS  OF  FIGURES. 

15.  Paraboloid. 

Rule. — The  distance  of  its  centre  from  the  base  along  its 
axis  is  ^  of  the  height  from  the  base. 


Fia.  100. 


16.  To  find  the  centre  of  gravity 
of  any  continuous  curved  line.  (Fig. 
100.) 

Ex. :  Let  ABC  be  the  given  curve ; 
bisect  it  at  B  ;  join  ab  and  bc,  and 
bisect  those  chords  at  the  points  d 
and  E  respectively  ;  set  off  fd  per- 
pendicular to  AB_,  and  EG  perpen- 
dicular to  BC  ;  make  fh  =  ^df  and 
GK==^GE,  and  join  hk  ;  bisect  Hk 
at  the  point  L,  vrhich  will  be  a  close 
approximation  to  the  position  of  the 
centre  of  gravity  of  the  curved  line 

ABC. 

Remark. — If  the  line  is  too  irregular  to  permit  its  two 
parts  to  be  regarded  as  flat  regular  curves,  it  should  be  divided 
into  four  or  eight  parts  as  required.  The  points  corresponding 
to  L  in  the  above  figure  are  found  separately  for  each  pair  of 
parts,  joined  in  pairs  and  bisected  ;  this  process  is  repeated 
until  only  one  point  remains,  this  being  the  required  centre 
of  gravity. 


eules   for   finding  the  moments  and   centres  of 
Figures. 

The  geometrical  moment  of  a  figure,  whether  a  line,  an 
area,  or  a  solid,  relatively  to  a  given  'plane  or  axis  is  the 
product  of  the  ^nagnittide  of  that  figure,  into  the  perpendictdar 
distance  of  its  centre  from  the  given  plane  or  axis,  and  is 
equal  to  the  sum  of  the  moments  of  all  its  parts  relatively  to 
the  same  plane. 

The  centre  of  an  area  is  determined  when  its  distance  from 
two  axes  in  the  plane  of  the  figure  is  known. 

The  centre  o^  a  figfure  of  three  dimensions  is  determined 


CENTRES  AND  MOMENTS  OF   FIGURES. 


68 


when  its  distance  from  tln-ee  planes  not  parallel  to  one  another 
is  known. 

1.  To  find  the  moinent  of  an  ir^'cgula?'  fignre  relatively  to  a 
given  plane  or  axis. 

Rule. — Divide  the  figui*e  into  parts  whose  centres  are  known ; 
rcultiply  the  magnitude  of  each  of  its  parts  into  the  perpendi- 
cular distance  of  its  centre  from  the  given  plane  or  axis  ;  dis- 
tinguish the  moments  into  positive  and  negative,  according  as 
the  centres  of  the  parts  lie  to  one  side  or  the  other  of  the  plane  : 
the  difference  of  the  two  sums  will  be  the  resultant  moment  of 
the  figure  relatively  to  the  given  plane  or  axis,  and  is  to  be 
regarded  as  positive  or  negative,  according  as  the  sum  of  the 
positive  or  negative  moments  is  the  greater. 

2.  To  find  the  pe7'pendicular  distance  of  t/ie  centre  of  an  irre- 
gular figure  from  a  given  plane  or  axis. 

Rule. — Divide  the  moment  of  that  figure  relatively  to  the 
given  plane  or  axis  by  its  magnitude  ;  the  quotient  will  be  the 
perpendicular  distance  of  its  centre  from  the  given  plane  or  axis. 

3.  To  find  the  centre  of  a  figure  consisting  of  two  j^a^'ts  whose 
centres  are  known.    (Fig.  101.) 

Rule.— Multiply  the  distance  between  the  two  known  cen- 
tres by  the  magnitude  of  either  of  the  parts,  and  divide  the 
product  by  the  magnitude  of  the  whole  figure  ;  the  quotient 
will  be  the  distance  of  the  centre  of  the  whole  figure  from  the 
centre  of  the  other  part,  the  centre  of  the  whole  figure  being 
in  the  straight  line  joining  the  centres  of  the  two  parts. 

Ex.:  Let  abcd  be  such  a  figure,  M  and  m 
the  magnitude  of  its  two  respective  parts,  M  +  m 
the  magnitude  of  the  whole  figure,  D  the  dis- 
tance between  the  centres  M  and  in  of  the  two 
parts,  and  c  the  centre  of  the  whole  figure. 


Fig.  101. 


MC  =  : 


M  X  D 

■  M  +  w' 


Fig.  102. 


4.  To  find  the  centre  of  any  j^lane  area  by  means  of  or di nates. 
(Fig.  102.) 

Let  ABC,  the  quadrant  of  a  circle,  be  such 
an  area ;  CB  the  base  line,  divided  into  a 
number  of  equal  parts  by  ordinates ;  AC  the 
transverse  axis  traversing  its  origin. 

1st.  Determine  the  peipendiciilar  distance 
of  the  centre  of  the  quadrant  from  the  trans- 
verse axis  in  the  following  manner : — 

Rule. — Multiply  each  ordinate  by  its  dis- 
tance from  the  transverse  axis ;  consider  the 
products  as  ordinates  of  a  new  curve  of  the  same  length  as  the 
given  figure  :  the  area  of  that  curve,  found  by  the  proper  rule, 
will  be  the  moment  of  tlie  figure  relatively  to  the  transverse 


^:tTm^^ 


G4 


CENTRES   AND   MOMENTS  Of   FIGURES. 


axis ;  this  moment,  divided  by  the  whole  area  of  the  figure,  will 
give  the  perpendicular  distance  of  its  centre  from  the  transverse 
axis. 

In  algebraical  symbols  the  moment  of  a  plane  figure  rela- 
tively to  its  transverse  axis,  and  found  by  the  above  rule,  is 
expressed  thus : — 

fxydx. 

Note. — In  practice  it  is  better  to  proceed  as  follows  : — Multiply 
the  ordinates  first  by  their  multipliers,  and  then  those  products 
by  the  number  of  intervals  from  the  origin  ;  take  the  sum  of 
those  products  and  multiply  it  by  ^rd  of  a  whole  interval 
squared,  if  Simpson's  first  rule  is  used,  by  fths  of  a  whole  inter- 
val squared,  if  Simpson's  second  rule  is  used,  and  so  on  for  the 
other  rules. 

Example. 


No.  of 
Intervals 


Ordinate? 


0 
1 
2 

3 

^ 

4 


16-0000 

1.5-4919 

13-8564 

12-4900 

10-5830 

9-3274 

7-7460 

5-5678 

0-0000 


Multi- 
pliers 


1 
4 

2 


Products 


16-0000 

61-9676 

20-7846 

24-9800 

7-93725 

9-3274 

3-8730 

5-5678 

0-0000 


Products  X  No.  of  Intervals 
from  Origin 


•00000 
61-9676 
41-5692 
62-4500 
23-81175 
30-31405 
13-5555 
20-87925 

•00000 


Interval  150-4376^ 


Interval  - 
3 


254-54735 


Approximate  area  =  200-58353Approx.  moment  =  1357-585 

Moment  1357-585  ^  c.nna   /"approximate  perpendicular  distance 
Area  200-5835  \   of  centre  from  the  transverse  axis. 

2tid.  Find  the  perpendicular  distance  of  its  centre  from  tlie 
base  line. 

KuLE. — Square  each  ordinate,  and  take  the  half -squares  as 
ordinates  for  a  new  curve  of  the  same  length  as  the  figure  ;  the 
area  of  that  curve,  found  by  the  proper  rule,  will  be  the  moment 
of  the  figure  relatively  to  the  base  line  :  this  moment,  divided 
by  the  w^hole  area  of  the  figure,  will  give  the  perpendicular 
distance  of  its  centre  from  the  base  line. 

In  algebraical  symbols  the  moment  of  a  plane  figure  rela- 
tively to  its  base  line,  foimd  by  the  above  rule,  is  expressed 

thus :  — 


/ 


^-Ldx. 
2 


CEiNTRES   AND   MOMENTS  OF   FIGURES. 
E.xamplc. 


65 


No,  of  Int^vals 

Ordiua'.es 

Half-squares 

JIultipliers 

rroducta 

1 
2 

4 

16-0000 
15-4919 
13-8564 
10-5830 
0-0000 

128-0000 

119-9995 

95-9999 

55-9999 

00000 

1 
4 

2 
4 
1 

128-0000 
479-9980 
191-9998 
223-9996 
00000 

Interval 
3 
Approximate  moment 

1023-9974 

=  1365-3298 

Moment  1365-3298 
Area20r0624~ 


=  6-796 


J  approximate  perpendicular  dis- 
\     tance  of  centre  from  base. 


Actual  moment  =  1365-3 
Actual  afea        ^201-0624 

5.  To  find  the  centre  of  a  plane  area  hounded  hj  a  curve  and 
two  radii  by  means  of  polar  co-ordinates.     (See  lig.  68.) 

1st.  Determine  the  perpendicular  distance  of  its  centre  from  a 
plane  ti'aversin/j  the  pole  and  at  inght  angles  to  one  of  the  hound- 
ing radii,  called  the  first  radius,  in  the folhiuing  manner: — 

KuLE. — Divide  the  angle  subtended  by  the  arc  into  a  conve- 
nient number  of  equiangular  intervals  by  means  of  radii ;  mea- 
sure the  lengths  of  the  radii  from  the  pole  to  the  arc,  and 
multiply  the  third  part  of  the  cube  of  each  of  them  by  the 
cosine  of  the  angle  which  they  respectively  make  with  the  first 
radius  ;  treat  these  products  by  one  of  the  rules  applicable  to 
finding  the  area  of  a  plane  curve  (the  only  difference  being  that 
the  common  interval  is  taken  in  circular  measure) ;  the  result 
will  be  the  moment  of  the  figure  relatively  to  the  plane  tra- 
versing the  pole :  this  moment,  divided  by  the  area  of  the 
figure,  will  give  the  perpendicular  distance  of  its  centre  from 
the  plane  traversing  the  pole. 

Exam2)le. 


>-o. 

of 

Radii 

Radii 

1 

12 

2 

12 

3 

12 

4 

12 

5 

12 

.jiCules  of  l?!idii 


576 
576 
576 
576 
576 


Anples! 
with 
First 

Radius 


0= 

5= 

10- 

15= 

20- 


Ckjsines    Products 


1-0000  576-0000 
-9962  573-8112 
-9848  567-2448 
-9659  556-3.584 
•9397  541-2672 


Simpson  s 
Multi- 
pliers 


Products 


Interval  in  circular  measure 
3 


576-0000 
2295-2448 
!ll34-4896 
2225-4336 
j  541-2672 

677F4352" 

-0291 


Moment  relatively  to  plane  traversing  pole  =  197-077864 
F 


66 


CENTRES  AND  MOMENTS  OF  FIGURES. 


Moment  197-077864 
Area  25-1327  ~ 


7-841  /perpendicular  distance  of  centre 
1     from  plane  traversing  pole. 

In  algebraical  symbols  the  moment,  as  here  found,  is  ex- 
pressed thus :  — 


f 


cos  ede. 


2nd.  Determine  the  Tnoinetit  of  the  figure  relaiively  to  tlie  first 
radius  precisely  in  the  same  ivay  as  in  the  foregoing  rule,  with  the 
exception  tluit  sines  must  he  used  in  the  itlace  of  cosines;  this 
moment,  divided  hy  the  area  of  the  figure,  will  give  the  perpen- 
dicular distance  of  its  centre  from  the  first  radius. 

Note. — It  is  usual,  in  practice,  to  defer  the  division  of  the 
cubes  of  the  radii  by  3  until  after  the  addition  of  the  products. 

Example. 


No. 

of 

Radii 

Radii 

1 

12 

2 

12 

3 

12 

4 

12 

5 

12 

Cubes  of  Radii 


576 
576 
576 

576 


Angles 
with 
First 

Radius 

Sines 

of 
Angles 

Products 

Simpson's 
Multi- 
pliers 

0° 

•0000 

-0000 

1 

5° 

•0872 

50-2272 

4 

10° 

•1736 

99-9936 

2 

15° 

•2588  1149-0688 

4 

20° 

-3420 

196-9920 

1 

Products 


•0000 
200-9088 
199-9972 
596-2752 
196-9920 


Interval  in  circular  measure 


1194-1732 
•0291 


Moment  relatively  to  first  radius  =    34*750440 

Moment  34-75044  f  perpendicular  distance  of  centre  from 

Area  25-1327     "  ^  ^«  \     first  radius. 

In  algebraical  symbols  the  moment  as  here  found  is  ex- 
pressed thus  : — 


/ 


-^  sin  QdQ. 


6.  To  find  the  perpendicular  distance  of  the  centre  of  a  solid, 
hounded  on  one  side  hy  a  curved  surface  (figs.  87  and  88),  from 
a  plane  perpendicular  to  a  given  axis  at  a  given  point. 

KuLE. — Proceed  as  in  Kule  4,  p.  63,  to  find  the  moment 
relatively  to  the  plane,  substituting  sectional  areas  for  breadths  : 
then  divide  the  moment  by  the  volume  (as  found  by  Kule  2,  p.  54) ; 
the  quotient  will  be  the  required  distance.  To  determine  the 
centre  completely,  find  its  distance  from  three  planes  no  two  of 
which  are  parallel. 


CENTRES   AND   MOMENTS  OF  FIGURES. 


67 


Fig.  103. 


7.  Having  the  itwment  and  centre  of  a  ff/urc  relativrbi  to  a 
given  2>lune^  to  find  the  nciv  moment  and  centre  of  the  fgiue  rela- 
tively to  the  same  plane  n-hcn  a  part  of  the  figure  is  shifted. 
(Fig.  103.) 

In  the  figure  wlk  let  c  be  its 
centre,  and  zz'  a  plane  with  respect 
to  which  the  moment  of  the  figure  is 
known  ;  suppose  the  part  avsm  to  ^^ 
be  transferred  to  the  new  position 
SNL,  so  as  to  alter  the  shape  of  the 
figure  from  wlk  to  mnk  ;  let  i 
be  the  original  and  H  the  new  cen-" 
tre  of  the  shifted  part :  then  the 
momeyit  of  the  p'gure  mnk  relatively 
to  the  plane  zz'  is  found  as.  follows  : — 

Rule.— Measure  the  distance,  perpendicular  to  the  plane  of 
moments,  between  the  centres  of  the  original  and  new  position 
of  the  shifted  part,  as  hd,  and  multiply  it  by  the  magnitude 
of  the  shifted  part ;  the  product  will  be  the  moment  required. 
The  new  2>osition  of  the  entire  fgure  is  then  found  by  the  following 
rule: — 

Rule. — Multiply  the  distance  between  the  centres  of  the 
original  and  new  position  of  the  shifted  part  by  the  magnitude 
of  that  part ;  that  product,  divided  by  the  magnitude  of  the 
whole  figure,  will  give  the  distance  the  centre  has  traversed  in 
the  direction  in  which  the  part  has  been  shifted,  and  in  a  plane 
parallel  to  a  line  joining  the  centres  of  the  original  and  new 
position  of  the  shifted  part,  as  from  c  to  c'  in  fig.  103. 

8,  To  find  the  centre  of  a  wedge-shaped  solid  (fig.  104)  by 
means  of  polar  co-ordinates. 

\st.  Determine  the  perpendicular  distance  of  its  centre  rela" 
tively  to  a  transverse  sectional  plane,  as  pab. 

Rule. — Divide       the  y\g.  io4,  ■ 

solid  by  a  number  of 
parallel  and  equidistant 
planes,  as  pab,  p,a,b,, 
PjAjB,,  tfcc. ;  then  mul- 
tiply each  sectional  area 
by  its  distance  from  the 
plane  pab  ;  treat  the 
products  as  though  they 
were  the  ordinates  of  a  curve  of  the  same  length  as  the  figure  ; 
the  area  of  that  curve,  found  by  the  proper  rule,  will  be  the 
moment  of  the  figure  relatively  to  the  plane  pab  :  that  moment, 
divided  by  the  volume  of  the  figure,  will  be  the  distance 
required. 


68  CENTRES   AND   MOMENTS  OF   FIGURES. 

2)id.  Determine  the  i^erpendicular  distance  of  its  centre  re- 
latively to  a  longitudinaL  plane  j^assing  through  its  edge,  as  MPM, 
l)crpendicular  to  the  first  radius,  pb. 

Rule. — Divide  the  figure  by  a  number  of  longitudinal 
planes  radiating  from  the  ed.^^'e  mpm  at  equiangular  intervals 
(as  pp^AA^,  PP4CC^,  PP4BB4) ;  also  divide  the  length  of  the  figure 
into  a  number  of  equal  intervals  by  ordinates,  and  treat  eacli 
of  the  longitudinal  planes  as  follows  :  — Measure  its  ordinates, 
take  the  third  part  of  their  cubes,  and  treat  those  quanti- 
ties as  if  they  were  ordinates  of  a  new  curve ;  that  is,  find  its 
area  by  one  of  Simpson's  rules  :  the  area  of  that  new  curve  is 
termed  the  moment  of  inertia  of  the  longitudinal  plane  in 
question.  Then  multiply  each  moment  of  inertia  of  the  several 
planes  by  the  cosine  of  the  angle  made  by  the  plane  to  which  it 
belongs  with  the  plane  pb,  and  treat  these  products  by  a  proper 
set  of  Simpson's  multipliers  ;  add  together  the  products,  and 
multiply  the  sum  by  \  of  the  common  angular  interval  in  cir- 
cular measure  if  Simpson's  first  rule  is  used,  and  by  |  if  Simp- 
son's second  rule  is  used.  The  result  will  be  the  moment  of 
the  figure  relatively  to  the  plane  MPM.  This  moment,  divided  by 
the  volume  of  the  figure,  will  be  the  distance  required. 

The  algebraical  expression  for  the  moment  as  found  in  this 
rule  is 


././ 


^.3 

-^  cos  Od.ndB, 


Srd.  Determine  the  2)e7'pendicular  distance  of  its  centre  re- 
lativehj  to  a  longitudinal  plane  passing  through  its  edge,  and  a 
radius  as  pp*bb*,  by  the  foregoing  rule,  with  the  exception  of 
multiplying  by  sines  instead  of  cosines. 

Note. — In  practice  it  is  usual  to  defer  the  division  of  the 
cubes  of  the  radii  by  3  until  after  the  addition  of  the  products. 

9.  To  find  the  centre  of  gravity  of  a  plane  area  contained 
between  two  consecutive  ordi7iates,  with  respect  to  the  near  end 
ordinate. 

RuuE. — To  the  sum  of  three  times  the  near  end  ordinate,  and 
ten  times  the  middle  ordinate,  subtract  the  far  end  ordinate,  and 
multiply     the     remainder     by     the  Fig.  105, 

square  of  the  common  interval.  The 
product,  divided  by  24,  will  be  the 
moment  about  the  near  end  ordinate,  p^ 
On  dividing  this  by  the  area,  the 
longitudinal  position  of  the  centre  of 
gravity  is  obtained. 

Ex.\   In  fig.   105  let  abc  be  the 
base,  and  ad,  be,  and  CF  the  ordi- 
nates.     Call    them    y^,   1/2?  ^^^  th  A" 
respectively,     and     let     the     common     interval     be     denoted 
by  h.      Then    the    moment    of    tJio    area    abed    about    the 


•h--i 


MOMENTS   OF    INERTIA    AND    RADII    OF    GYRATION.       69 

near  end  ordinate  ad  is  equal  to  ^^-^'-t.l^^2":2^«)  ^J^'.  If  this  be 

24 
divided  by  the  area  of  abed  (see  p.  46),  the  quotient  will  be  the 
distance  of  the  C.G.  from  ad. 

For  an  example,  let  the  ordinates  be  62,  8-5,  and  94  feet, 
and  the  common  interval  12  feet. 


No.  of 
Ordi- 
nates 

1 

2 
3 

Ordinates 

6-2 
8-5 
9-4 

Multipliers 
for  Area 

5 

8 

-1 

(Interval 

Products 

Ordinates 

Multipliers 

for 
Moments 

Products 

310 
68  0 
-9-4 

89-6 
)-       1 

6-2 

8-5 
9-4 

Momen 

3 

10 
~1 

(Interval) 

18-6 
85  0 
-9-4 

94-2 
.^-       6 

Area  of  portion 
between  1  and 

12 

ncluded " 

12         .  J 

.  =89-6 

24 
t  about  1  =  565-2 

Moment  5652 


Area 


89-6 


=  6-308 


Perpendicular  distance  of  centre  of 
portion  included  between  Nos.  1 
and  2,  from  No.  1  ordinate. 


Note.— When  the  moment  of  the  area  is  required  about  the 
middle  ordinate,   the  above  multipliers    should   be  changed   to 
Tyi  +  6?/2 -j/3  ^  ^^2^ 


1 ;    so  that  moment  = 


24 


Moments   of   Inertia    a.vd  Radii  op  Gyration. 

1.  To  find  the  moment  of  inertia  of  a  body  about  a  given 
axis. 

Rule. — Conceive  the  body  to  be  divided  into  an  indefinitely 
great  number  of  small  parts;  multiply  the  mass  (or  area)  of 
each  of  these  small  parts  into  the  square  of  its  perpendicular 
distance  from  the  given  axis  :  the  sum  of  all  these  products  as 
obtained  will  be  the  moment  of  the  body  about  the  given  axis. 

2.  To  find  the  square  of  the  radius  of  gyrnlion  of  a  body 
about  a  given  axis. 

Rule. — Divide  the  moment  of  inertia  of  the  body  relatively 
to  the  givfM)  axis  by  the  mass  (or  area)  of  the  body. 


70  MOMENTS    OF    INERTIA.. 

3.  Given  the  moment  of  ineHla  of  a  body  about  an  axu 
traversing  its  centre  of  gravity  in  a  given  direction,  to  find  its 
moment  (f  inertia  about  another  axis  parallel  to  the  first. 

Rule. — Multiply  the  mass  (or  area)  of  the  body  by  the 
square  of  the  perpendicular  distance  between  the  two  axes,  and 
to  the  product  add  the  given  moment  of  inertia. 

4.  Given  the  separate  moments  of  ineHia  of  a  set  of  bodies 
about  pai'allel  axes  traversing  their  several  centres  of  gravity,  to 
find  the  moment  of  inertia  of  these  bodies  about  a  com,vwn  axis 
parallel  to  their  separate  axes. 

Rule. — Multiply  the  mass  (or  area)  of  each  body  by  the 
square  of  the  perpendicular  distance  of  its  centre  of  gravity 
from  the  common  axis ;  the  sum  of  all  these  products,  together 
with  all  the  separate  moments  of  inertia,  will  be  the  combined 
moment  of  inertia. 

5.  Given  the  square  of  the  radius  of  gyration  of  a  body  about 
an  axis  traversing  its  centre  in  a  given  direction,  to  find  the 
square  of  the  radivs  of  gyration  abont  another  axis  parallel  to  the 
first. 

Rule. — Square  the  perpendicular  distance  between  the  two 
axes,  and  add  the  product  to  the  given  square  of  the  radius  of 
gyration. 

6.  To  find  the  moment  of  ineHia  of  a  plane  area,  bounded  on 
one  side  by  a  curve  (see  fig.  102),  relatively  to  its  base  line. 

Rule. — Divide  the  base  line  into  a  suitable  number  of  equal 
intervals,  and  measure  ordinates  at  the  points  of  division;  take 
the  third  part  of  the  cube  of  each  of  these  ordinates,  and  treat 
those  quantities  so  computed  as  the  ordinates  of  a  new  curve  : 
the  area  of  that  new  curve,  foimd  by  the  proper  rule,  will  be  the 
moment  of  inertia  required.  In  algebraical  symbols  the  above 
rule  is  expressed  thus  : — 

fta.. 

Note. — When  the  moment  of  inertia  is  required  as  a  whole, 
and  not  in  separate  parts,  it  is  usual  to  postpone  the  division  of 
the  cubes  till  the  end  of  the  calculation. 

7.  To  find  the  moment  of  inertia  of  a  plane  area,  bounded  on 
one  side  by  a  cnrr:e,  relatively  to  one  of  its  ordinates. 

Rule. — Multiply  each  ordinate  by  its  proper  multiplier,  ac- 
cording to  one  of  the  rules  for  finding  the  area  of  such  figures ;  then 
multiply  each  of  the  products  by  the  square  of  the  number  of 
whole  intervals  that  the  ordinate  in  question  is  distant  from  the 


MOMENTS    OF   INERTIA.  71 

ordinate  taken  as  the  axis  of  moments  :  the  sum  of  these  pro- 
ducts, multiplied  by  |  or  |  the  cube  of  a  whole  interval,  accord- 
ing as  Simpson's  first  or  second  rule  is  used,  will  be  the  moment 
of  inertia  required. 

In  algebraical  symbols  this  rule  is  expressed  thus  :— 
fx-ij(hv. 

Example  I. 

Calculation  of  Moment  op  Inertia  op  the  Quadrant 
OP  A  Circle  Relatively  to  the  Base  Line. 


No.  of  Intervals 

Ordinates 

Cubes  of  Ordinates 
3 

Multipliers 

Products 

0 
1 
2 
2i 

16-00 
15-49 
13-86 
12-49 

1365-33 

1238-89 

887-50 

649-48 

1 
4 

2 

1365-33 

4955-56 

1331-25 

1298-96 

296-07 

270-72 

77-58 

57-29 

0-00 

3 

n 

3f 
4 

10-58 
9-33 
7-75 
5-57 
000 

394-76 

270-72 

155-16 

57-29 

0  00 

1 

r 
i 

Interval 
3 

9652-76 

12870-34 

Example  II. 

Calculation 

I  OF  THE  M< 

3MENT  OF INERI 

riA  OF  THE 

Quadrant 

No.  of 
Interrals 

Ordinates 

Multipliers 

Products 

Squares  of  Nos. 
of  Intervals 

Products 

0 

160000 

1 

16-0000 

000 

000 

1 

15-4919 

4 

61-9676 

1-00 

61-9679 

2 

13-8564 

u 

20-7846 

4-00 

83-1384 

n 

12-4900 

2 

24-9800 

6-25 

156-1250 

3 

10-5830 

1 

7-93725 

9-00 

71-4353 

3i 

9-3274 

1 

9-3274 

10-5625 

98-5207 

3^ 

7-7460 

h 

3-8730 

12-2500 

47-4443 

3| 

5-5678 

1 

5-5678 

14-0625 

78-2972 

4 

S)0000 

i 

00000 

160000 

00000 

A 

pproxima 

te  momei 

Interval 
3 

It  of  inert  ia  = 

,    596-9288 

12734-4810 

72 


MOMENTS    OF   INERTIA. 


Definition. — If  a  body  be  conoeivcd  divided  into  an  infinite 
number  of  parts,  and  the  mass  (or  area)  of  each  part  be 
multiplied  by  the  square  of  its  distance  from  a  fixed  point, 
the  sum  of  all  these  products  is  termed  the  polar  moment  of 
inertia  about  the  point. 

8.  To  find  the  'polar  moment  of  inertia  of  a  plane  area 
about  a  ^oint. 

i  (I)  EuLE. — At  equal  angular  intervals  sufficient  to  in- 
clude the  whole  area,  draw  radii  from  tlie  point  to  the  peri^ 
meter.  Treat  the  fourth  power  of  these  radii  as  the  ordinates 
of  a  new  curve  having  a  common  interval  equal  to  the  angular 
interval  between  consecutive  radii  expressed  in  circular 
measure.  One  quarter  of  the  area  of  this  curve,  found  by  the 
proper  rule,  is  the  polar  moment  of  inertia  required. 
!  Example. — Find  the  polar  moment  of  inertia  of  a  semi- 
circle of  5  feet  radius  about  one  end  of  the  diameter, 
i  The  polar  radii  at  an  angular  interval  of  15"  are  1000, 
9-66,  8-66,   7-07,  5-00,   2'59  feet. 


No. 

Radius 

(Radius)^ 

Multiplier 

Product 

1 

10-00 

10,000 

1 

10,000 

2 

9-06 

8,735 

4 

34,940 

3 

8-66 

5,624 

2 

11,248 

4 

7-07 

2,498 

4 

9,992 

5 

5-00 

625 

2 

1,250 

6 

2-59 

45 

4 

180 

7 

— 

— 

1 



Common  i 

nterval  =  -2618 

67,610 

■ 

X  t^  X  J  X  -2618 

Polar  moment  of  inertia  =  1,475 

(II)  Rule. — If  the  moments  of  inertia  about  two  perpen- 
dicular axes  through  the  point  are  known,  their  sum  is  equal 
to  the  polar  moment  of  inertia  about  the  point. 

definitions. — The  product  of  inertia  of  an  area  about  two 
perpendicular  axes  is  the  algebraic  6um  of  each  element  of  area 
multiplied  by  the  product  of  its  co-ordinates  with  reference 
to  the  two  axes.  In  the  first  and  third  quadrants  the  product 
of  inertia  is  positive  ;  in  the  second  and  fourth  quadrants 
it  is  negative. 

The  principal  axes  of  inertia  through  a  point  are  those 
axes  about  which  the  product  of  inertia  is  zero. 

9.  Given  the  moments  and  products  of  inertia  about  iivo 
perpendicular  uxes,  to  find  the  corresjjonding  qtmntities  about 
any  two  other  perpendicular  axes. 

llULE. — It'  OX,  oy  (fig.  106)  are  the  axes,  X  and  Y  the  moments 
of  inertia  about  them,  and  P  their  product,  the  moments  and 


MOMENTS    OF    INERTIA.  73 

product  of  inertia  about  Ox,  Oy'  (denoted  by  x',  y',  and  p'),  making 
a  positive  angle  6  with  the  original  axes,  are  given  by  the 
following  formulae : — 

x'  =  X  cos"  6  +  \  sin"  0  -  2p  sin  6  cos  0, 
y'  =  X  sin-  B  +  Y  cos^  0  +  2p  sin  9  cos  0, 

80  that  x'  +  y'  =  X  +  Y  ;  and 

p'  =  p  cos  20-  ^Y-X)  sin  20. 
Note. — If    ox    and    oy  are   principal  axes,   P  =  0,  and  the 
formulse  become 

x'  =  X  cos-  0  +  Y  sin'  0  ;  y'  =  X  sin-  0  +  Y  cos"  0  ; 
P'  =  -  §  (y  -  x)  sin  20. 


9    ^ 


If  an  ellipse  (fig.  106)  be  drawn  having  its  principal 
axes  ox,  oy  along  the  principal  axes  of  inertia,  and  of 
magnitude  oa,  ob  equal  to  radii  of  gyration  about  oy  and  Ox 
respectively,  the  radius  of  gyration  about  any  other  axis  Oa;' 
is  represented  by  the  perpendicular  OM  drawn  to  that  tangent 
of  the  ellipse  which  is  parallel  to  the  axis  Ox'  ;  the  moment 
of  inertia  about  Ox'  is  proportional  to  the  square  of  OM,  or 
equally  inversely  proportional  to  the  square  of  the  radius  op 
along  ox' .  The  product  of  inertia  about  ox',  oy'  is  similarly 
represented  by  the  product  of  Oil  and  MQ,  where  OQ  is  con- 
jugate to  OP. 

10.  Given  the  moments  and  product  of  inertia  about  two 
perpendicttlar  axes,  to  find  the  principal  moments  and  axes 
of  inertia.  • 

Rule. — If  x,  y,  and  P  are  the  moments  and  product  of 
inertia  resx>ectively  for  the  axes  ox,  oy,  the  angle  0  (reckonecl 
positively)  made  by  the  principal  axes  ox',  Oy' ,  with  the 
original  axes  is  given  by  the  formula — 

f      o »         2P 
tan  2t'  =-•  ^^-^ 


74  MOMENTS    OF    INERTIA, 

The  magnitudes  of  the  principal  moments  of  inertia  x',  Y' 
about  ox',  ot/',  are  given  by — 


assuming  x'   to  be  the  least  and  y'   the  greatest  moment  of 
inertia. 

11.  Given  the  moments  of  inertia  about  three  axes,  two 
perpendicular  and  one  bisecting  the  angle  betiveen  them,  to 
find  the  principal  moments  and  axes  of  inertia. 

Rule. — If  x,  y,  are  the  moments  of  inertia  about  the  axes 
ox,  oy,  and  z  that  about  an  axis  bisecting  the  angle  yox,  tlie 
angle  Q  (reckoned  positively)  made  by  the  principal  axes 
Ox't  Oy' ,  with  the  original  axes  is  given  by  the  formula — 

Y  +  X  -  2Z 
tan  20  =  — ^— Y~~ 

The  magnitudes  of  the  principal  moments  of  inertia  \' ,  Y' 
about  ox\  oy' ,  are  given  by — 


^±---Vz^-(Y>X)  +  ^4^' 


y'  =l±?+\/z'-Z(Y  +  x)  + 


Y^  +  X^ 


Note.—^inae  X  +  Y  =  x'  +  y',  the  sum  of  the  moments  of 
inertia  about  any  two  perpendicular  axes  is  constant. 

If  the  area  has  an  axis  of  symmetry,  the  principal  axes 
are  along  and  perpendicular  to  this  axis. 

Ex. — An  unequal-sided  parallelogram  is  formed  of  two  right- 
angled  Isosceles  triangles  of  1  inch  side.  Find  the  principal 
moments  and  axes  of  inertia. 

Take  ox  parallel  to  the  shorter  sides,  and  oz  perpendicular 
to  the  longer  sides.     Then.x  =  ^\    Y  =  J  ;    Z  =  i^\. 

By  the  formulae  above  tan  20  =  -  2 ; 

0  =  -  32°  or  58°,  the  former  corresponding  to  the  least  moment 
of  inertia. 

Greatest  i  or  y'  =  J  +  -gj  =  -218. 
Least      I  or  x'  =  J  -  2^  =  -032. 


Table  of  Squares  op  Eadh  op  Gyration  op  a  fkw  Spectaij  Figures. 


body 


Rectangle ;  sides  a  and  b 
Square ;  side  a 


Triangle ;    sides  a,   h,   e ; 
heights  a',  b',  c' 

Equilateral  triangle ;  height  d 


Trapezoid ;  height  h,  parallel 
sides  a  and  b 

Trapezoid  with  two  right 
angles ;  parallel  sides  a  and 
b,  perpendicular  side  h 

Circle ;  diameter  a 
Ellipse  :  diameters  a,  b 


Common  parabola;  height  a, 
base  b  perpendicular  to  axis 


Sphere ;  radius  r 


Spherical  shell ;  external  and 
internal  radii  ri  and  ra 

Ellipsoid  of  revolution ;  trans- 
verse semi-axis  r 

Ellipsoid :  semi-axes  a,  b,  o 


Circular  cylinder;  radius  r, 
length  2a 


Hollow  circular  cylinder ; 
radius — external  ri,  internal 
r2 ;  length  2a 


Baiiptic  cylinder;    semi-axes 
b,  c,  length  2a 


Tone :    height    7i,   radius   of 
"  Else  r 


side  a 

axis  through  C.G.  parallel 
to  side  a 

any  axis  through  C.Q. 

side  a 

axis  through  C.G.  parallel 
to  side  a 


any  axis  through  O.Q. 


/  . 


side  a 


axis  through  C.Q.  parallel 


I  axis 
[    to 


side  a 


Bide  h 


axis  through  C.G.  parallel 
i     to  side  h 
i  diameter 

1  centre  (polar) 

diameter  a 

f  axis  of  parabola 

j  base  b 

I  axis  through  C.G.  parallel 
I.    to  base  b 

diameter 
centre  (polar) 
j  diameter 

!•         axis  of  revolution 

axis  2a 

{longitudinal  axis 
transverse    diameter 
through  C.G. 

C        longitudinal  axis 

I  transverse    diameter 
L    through  C.G. 

f        longitudinal  axis 

I  transverse  axis  2b  through 
I.    C.G. 

'         longitudinal  axis 

transverse    axis    through 

C.G. 
transverse    axis    through 

base 

plane  of  base 


8 

b» 
12 

«i 
12 

«:? 

6 

18 

4! 

13 

^    q+Sb 

6  •  a+b 

Ti?    a'^  +  iab  +  b^ 

18*      (a-l-b)2 
I(a2-l-b2) 
ai  +  2a%+2abS+hi 


18  (a+b)^ 

a2/16 

a2/8 

ba 

16 

2Z>2/5 

8a2/35 

12a2/175 

5" 

8r5| 
6 

2(rig-r2g) 

6  (ri8-r28) 

2ra 

5 

b2  +  C« 

6 


4      3 

ri^-f-ra" 
2 

4       """S 

b^  +  c'i 


4 

_c2    oa 

4'^8 

To'*' 

j-3 

10    20 

10 


Mcment  of  Inertia  =  square  of  radius  of  gyration  x  mass  (or  area)  of  the  flKure. 


76 


MECHANICAL   PRINCIPLES 


MECHANICA^L   PRINCIPLES. 

Resultant  and  Resolution  of  Forces. 
1.  To  find  the  resultant  of  two  forces  acting  through  one  jMnnt 
hut  not  in  the  same  direction.     (Fig.  107.) 


Let  AB,  AC  represent  the  two 
forces  P  and  Q  acting  through  the 
point  A;  complete  the  parallelo- 
gram ABCD  :  then  its  diagonal  ad 
will  represent  in  magnitude  and 
direction  the  resultant  of  the  two 
forces  p  and  Q. 

R  =  resultant.  0  =  an 

a  =  angle  R  makes  with  q.     )8 


Fig.  107. 


le  P  makes  with  Q. 
angle  R  makes  with  p. 


R=  a/p2  +  Q'-  +  2.p.q.cos0; 
,P 


sin  a  =  sin  6- 


sin  )3  =  sin  d 


R 


Fig.  108. 


2.  To  find  the  resultant  of  any  number  of  forces  acting  in  the 
same  plane  and  tJi?'ough  one  point  but  not  in  the  same  direction. 
(Fig.  108.) 

Let  p,  P,,  P.^,  P3  be  the  forces 
acting  through  the  point  of 
application  o ;  commence  at  o 
and  construct  a  chain  of  lines 
OP,  PA,  AB,  BC,  representing  the 
forces  in  magnitude  and  paral- 
lel to  them  ;  let  C  be  the  end 
of  the  chain :  then  a  line  R 
joining  oc  will  represent  in 
magnitude  and  direction  the 
resultant  of  the  forces  p,  p„  Pg, 
and  Pg. 

Note. — This  geometrical  pro- 
blem is  true  whether  the  forces 
act  in  the  same  or  in  different  planes. 
R  =  resultant. 
0  =  angle  made  by  R  with  a  fixed  axis  ox, 

a,  «„  «.,,  See.  =  angles  made  by  the  forces  P,  P,,  P.^,  &;c.,  with  ox. 

5x  =  sum  of  the  series  of  P  .  cos  a  +  P, .  cos  «,  +  Pg .  cos  a.^,  ko. 

2y  =  sum  of  the  series  of  p  .  sin  a  +  p, .  sin  a,  +  Po .  sin  a.,,  kc. 

R.cos0  =  2x.     R: 


R  .  sm  d  =  2y. 


(2xy  +  (2Y)"^ 

tan  e^ 

2y 
2x 

cos  e  = 

2x 

R 

sin  6=: 

5y 

R* 

MECHANICAL   PRINCIPLES  77 

3.  To  find  the   resultant  of  three  forces  acting  through  oiie 
point  and  making  right  angles  with'one  another.     (Fig.  109.) 

Fig.  109.  Let   OA,  OB,  OC  represent  in  magnitude 

and  direction  the  forces  x,  Y,  z  acting  through 
one  point  o  ;  complete  the  rectangular  solid 
AEFB  :  then  its  diagonal  OG  will  rejtrcsent 
in  magnitude  and  direction  the  resultant 
of  the  forces  x,  Y,  z. 
R  =  resultant, 
a,  3,  7  =  the  angles  R  makes  with  x,  Y,  z, 
respectively. 


Y  =  R  .  cos  j8.      R  =  A/'X*  +  Y'^  +  z2. 
Z  =  R  .  cos  7.      X  =  R  .  COS  O. 

4.  To  find  the  resultant  of  amj  number  offoi'ces  acting  through 
one  point  in  different  directions  ayid  not  in  the  same  plane. 

Let  p,  p„  P2,  &c.,  be  the  forces  o,  3,7 ;  o„  /8„  7,  ;  O2,  /3.^,  y^,  the 
angles  their  directions  make  with  three  axes  jmssing  through 
the  point  of  application  and  making  right  angles  with  one 
another. 


R  =  resultant. 

2x  =  p  .  cos  a  +  p,  .  cos 

a,  -f  P2  .  COS  a^  +  &c. 

2y  =  p  .  cos  )8  +  P,  .  cos 

3,  +  P2  .  cos  i8.^  +  &c. 

2z  =  p  .  cos  7  +  p,  .cos 

7,  +  Pj  .  cos  72  +  &c. 

R  = 

^/(2x)2  +  (2Y)•^+(2z)^ 

cos  a  = 

2x 

R 

cos  j8- 

2y 

R 

cos  7  = 

2z 
r' 

N.B.  Cosines  of  obtuse  angles  are  negative. 
Note. — P  cos  a,  p  cos  3,  and  P  cos  7  are  termed  the  components 
>f  the  forces  in  the  directions  of  x,  Y,  and  z  respectively.  The 
omponents  of  the  resultant  are  obtained  by  adding  (allowing  for 
jign)  the  components  of  the  several  forces  in  their  respective 
Ikections. 

Parallel  Forces. 
A  cou/ple  consists  of  two  equal   forces,  as  p  and   Q   (see 
If.   110),  acting  in  parallel  and  opposite  directions  to  one 
another,    and    is    termed    a    right-    or    left-handed 
Fig.  110.      couple,  according  to  whether  the  forces  tend  to  turn 
in  a  clockwise  direction  or  the  reverse. 

The  moment  of  a  couple  is  the  product  of  either 

the   forces    into   the   perpendicular   distance   ab 

between  the  lines  of  direction  of  the  forces.     The 

distance    AB    is    termed    the    arm    or  lever  of    the 

couple. 


-   —      coi 

hin 


78 


MECH\NIOAL   PRINCIPLES. 


Fig.  111. 


Fi3. 112. 


5.  To  find  the  resultant  moment  of  any  number  of  couples 
acting  upon  a  body  in  the  same  or  parallel  planes. 

Edle. — x\d(i  together  tJie  moments  of  tl:e  right-  and  left- 
handed  couples  separately  ;  the  difference  between  the  two 
sums  will  be  the  resultant  moment,  which  will  be  right-  or 
left-handed,  according  to  which  sum  is  the  greater. 

6.  To  find,  the  resultant  of  two  parallel  forces.  (Fig.  Ill 
and  112.) 

The  magnitude  of  the  resultant  of  two  parallel  forces  is 
their  sura  of  difference,  according  to  whether  they  act  in  the 
same  or  contrary  directions. 

Let  fig.  Ill  represent  a 
case  in  which  the  two  forces 
act  in  the  same  direction,  and 
%.  112  a  case  in  which  the 
components  act  in  opposite 
directions. 

Let  AB  and  CD  represent 
two  forces  ;  join  ad  and  CB, 
cutting  each  other  in  E;  in  da 
(produced  in  fig.  112)  take  df 
=BA  ;  through  F  draw  a  line 
parallel  to  the  components  ; 
this  will  be  the  line  of  the 
resultant,  and  if  two  lines  dg 
and  AH  be  drawn  parallel  to 
BC,  cutting  the  line  of  action 
of  the  resultant  in  0  and  H, 
GH  will  represent  the  magni- 
tude of  the  resultant. 
Or,  numerically,  the  line  of  action  of  the  resultant  is 
obtained  by  adding  (allowing  for  sign)  the  moments  of  the 
two  forces  about  any  (point,  this  being  equal  to  the  moment 
of  the  resultant  ;  the  perpendicular  distance  of  the  line  of 
action  from  the  point  is  obtained  by  dividing  this  moment  by 
the  magnitude  of  the  resultant. 


AF 


DC.  AD 


DF  =- 


AB.  AD 


GH  GH 

7.  To  find  the  resultant  of  any  number  of  parallel  forces. 
Rule. — Take  the  sum  of  all  those  forces  which  act  in  ont 

direction,  and  distinguish  them  as  positive  ;  tlien  take  the  sum 
of  all  the  other  forces  which  act  in  the  contrary  direction,  and 
distinguish  them  as  negative.  The  direction  of  the  resultant 
(positive  or  negative)  will  be  in  that  of  the  greater  of  these 
two  sums,  and  its  magnitude  will  be  the  difference  between 
them. 

8.  To  find  the  position  of  the  resultant  of  any  number  of 
parallel  forces  when  they  act  in  tivo  contrary  directions. 

Rule. — 1st.  Multiply  each  force  by  its  perpendicular  dis- 
tance from  an  assumed  axb  in  a  plane  perpendicular  to  the 


MECHANICAL   PRINCIPLES.  79 

linos  of  action  of  the  forces  ;  distinguish  those  momenta  into 
right-  and  left-handed,  and  take  their  resultant,  which  divide 
by  the  resultant  force  :  the  quotient  v.-ill  be  the  perpendicular 
distance  of  that  force  from  tiie  assumed  axis, 

2nd.  Find  by  a  similar  process  the  perpendicular  distance 
of  the  resultant  force  from  another  axis  perpendicular  to  the 
first  and  in  the  same  plane. 

9.  To  find  the  resultant  of  any  number  of  couples  not 
necessarily   in   a  plane. 

Two  couples  of  equal  moments  in  the  same  or  in  parallel 
planes  are  equivalent  to  one  another,  whatever  the  magnitudes 
and  positions  of  the  forces  composing  the  couples  may  be. 
A  couple  is  therefore  conveniently  represented  by  a  lino 
perpendicular  to  its  plane,  and  of  length  proportional  to 
its  moment  ;  usually  the  direction  of  the  lino  is  taken  so  that 
its  relation  to  the  direction  of  the  couple  is  the  same  as  that 
between  the  travel  and  the  rotation  of  a  right-handed  screw. 
Note  that  any  two  parallel  linos  of  the  same  magnitude  and 
sense  represent  the  same  couple. 

Rule. — Replace  tlie  couples  by  lines  as  above,  giving  them 
their  correct  magnitudes  and  direction,  and  treat  these  a? 
forces  through  a  point  by  Rule  4.  The  resultant  gives  the 
magnitude  and  direction  of  the  resultant  couple. 

10.  To  find  the  resultant  of  any  number  of  forces  in  a 
plane. 

Rule. — Treat  them  as  forces  through  any  fixed  point  by 
Rule  2,  and  find  their  resultant.  Calculate  also  the  moment  of 
each  force  about*^the  point,  and  add  them  together  allowing 
for  the  sign  of  each.  The  resultant  moment  divided  by  the 
magnitude  of  the  resultant  force  gives  the  perpendiculai: 
distance  of  its  line  of  action  from  the  point. 

Definition. — The  moment  of  a  force  about  a  line  that  it 
does  not  meet  is  the  product  of  the  component  of  the  force 
perpendicular  to  the  line  with  the  shortest  distance  between 
the  line  and  the  line  of  action  of  the  force. 

11.  To  fifid  the  resultant  of  any  number  of  forces,  not  in 
one  place. 

Rule. — Resolve  the  forces  parallel  to  three  perpendicular 
axes  as  in  Rule  4,  and  find  the  magnitude  and  direction  of  their 
resultant  E.  Calculate  the  moments  of  each  component  about 
the  three  axes,  and  treating  these  as  couples  find  the  resultant 
couple  F  by  Rule  9.  Resolve  this  couple  into  couples  a 
parallel  to  the  force  R,  and  h  perpendicular  to  B.  Resolve  the 
couple  H  into  2  foroas  E^,  E2,  of  which  Ej^  is  equal  and  opposite 
to  R,  while  Rg  is  equal  and  parallel  to  R  ;  find  the  position  of 
Ro  (not  in  plane  of  figure).  Then  the  final  resultant  is  equal 
to  the  force  R2  combined  with  the  couple  a  (since  R  and  Rj^ 
neutralize).  The  combination  of  a  force  in  and  a  couple 
jibout  tlie  same  line  is  termed  a  wrench. 


80  CENTRE    OF    GRAVITY    OF    BODIES. 

CENTRE  OF  GRAVITY. 

1.  To  Jincl  the  moment  of  a  hody's  n-cight  relativehj  to  a  given 
pl/ine. 

Rule. — Multijjly  the  w^eight  of  the  body  by  the  perpen- 
dicular distance  of  its  centre  of  gravity  from  the  given  plane. 

2.  To  find  the  common  centre  of  gravity  of  a  set  of  detached 
bodies  relatively  to  a  given  plane. 

KuLE.-— Find  their  several  moments  relatively  to  a  fixed 
plane  ;  take  the  algebraical'  sum  or  resultant  of  those  moments 
and  divide  it  by  the  total  sum  of  all  the  weights  :  the  quotient 
will  be  the  perpendicular  distance  of  the  common  centre  of 
gravity  from  the  given  plane. 

Note. — When  the  moments  of  some  of  the  weights  lie  on 
one  side  of  the  plane,  and  some  on  the  other,  they  must  be  dis- 
tinguished into  positive  and  negative  moments,  according  to  the 
side  of  the  plane  on  which  they  lie,  and  the  difference  between 
the  two  sums  of  the  positive  and  negative  moments  will  be  the 
resultant  moment.  The  sign  of  the  resultant  will  show  on  which 
side  the  common  centre  of  gravity  lies. 

Let  71',  m\  W-,  &c.  =  the  several  weights. 

rZ,  d\  d\  &c.  =  the  several  perpendicular  distances  of  the 
centres  of  gravity  of  w,  w',  w-,  &c.,  from  the  plane  of  moments. 

D  =  the  perpendicular  distance  of  their  common  centre  of 
gravity  from  the  plane  of  moments. 

_  w</  +  w^d^  +  rv'^d^  +  &c. 
?y  + w'  +w^  +  &c. 

3.  Ih  find  the  centre  of  gravity  of  a  body  consisting  of  parts 
of  unequal  heaviness. 

Rule. — Find  separately  the  centre  of  gravity  of  these  several 
parts,  and  then  treat  them  as  detached  weights  by  the  foregoing 
rule. 

4.  To  find  the  distance  through  which  the  common  centre  of 
gravity  of  a  set  of  detached  weights  moves  when  one  of  those  weights 
is  shifted  into  a  new  position. 

Rule.— multiply  the  weight  moved  by  the  distance  through 
which  its  centre  of  gravity  is  shifted  ;  divide  the  product  by  the 
sum  total  of  the  weights :  the  quotient  will  be  the  distance 
through  which  the  common  centre  of  gravity  has  moved  in  a 
line  parallel  to  that  in  which  the  weight  w^as  shifted. 

Let  w  =  weight  shifted. 

<Z  =  distance  through  which  w  was  moved. 

w  =  sum  total  of  weights. 

D=^  distance  through  which  the  common  centre  of  gravity 
has  moved  in  a  line  parallel  to  that  in  which  the  shifted  weight 
was  moved. 

^     wd        J     DW 


MOTION.  81 

MOTION. 

Velocity. 

The  speed  of  a  body  or  of  a  point  within  a  tody  is  the  distance 
travelled  in  an  infinifceisimal  space  of  time  divided  by  that 
time.  The  velocity  of  the  body  takef3  also  into  acooimt  the 
direction  in  which  the  body  is  moving  and  is  completely, 
represented  by  a  line  drawn  in  the  direction  of  motion,  whoso 
length  represents  to  scale  the  speed. 

Composition  of  velocities. — To  combine  several  velocities 
impressed  simultaneously  upon  a  body,  if  op,  op^,  0P2,  0P3 
(fig.  108,  p.  76)  represent  the  component  velocities,  draw 
PA  parallel  and  equal  to  op,  ab,  and  bg  parallel  and  equall 
respectively  to  0P2  and  0P3.  00  is  the  resultant  velocity  of 
the  body.  Similarly  the  resultant  velocity  oc  may  be  resolved 
into  two  component  velocities  in  any  required  directions  x  and 
Y  by  drawing  lines  from  od,  do  ptarallel  to  x  and  Y  ;  the 
lengths  od,  do  represent  the  magnitudes  of  the  component 
velocities. 

Exarntple. — If  a  boat  is  propelled  at  a  speed  and  in  a 
direction  represented  by  AC  (fig.  107,  p.  76)  in  a  stream  whoso 
velocity  is  represented  by  ab,  the  resultant  velocity  of  the 
boat  is  represented  by  ad.  To  combine  any  number  of 
velocities  analytically,  resolve  each  along  three  axes  at  right 
angles  (or  two  If  all  the  velocities  are  in  one  plane)  by 
multiplying  each  velocity  by  the  cosino  of  the  angle  which  it 
makes  with  the  axis  ;  add,  allowing  for  sign,  the  components 
along  each  direction.  The  sums  are  tho  components  of  tho 
resultant  velocity  in  the  three  directions,  wMch  may  bo  com- 
pounded as  above.  E.g.,  if  v^,  V2,  v^,  •  .  .  are  the  velocities 
making  angles  oi,  oo,  og,  .  .  .  with  the  axis  Ox,  0i,  jSo,  P3  -  -  •  with 
the  axis  oy,  and  71,  72,  73,  .  .  .  v»'ith  the  axis  oz,  the  components 
p,  Q,  R,  of  the  resultant  along  ox,  oy,  oz,  are  given  by — 
P  =  ^1  cos  ai  -\-  Vi  cos  0-2  +  Vs  cos  03  +  ... 
Q  =  vi  cos  Pi  +  V2  COS  /Sa  +  vs  cos  )33  +  .  .  , 

E  =  Vl  COS  7i  +  V2  COS  72  +  Vs  SOS  78-1-    •  .  • 

The  resultant  s  is  given  by  s^  =  P^  +  Q^  +  R- ;  and  it  makes 
angles  A,  B,  c,  with  the  axis,  given  by — 

P  Q  R 

COS  A  =  -  :  COS  B  =  -  ;  cos  c  =  — 
S  S  b 

Velocity   diagram  for  a  linJced  iTtechanism. — To   find   the 

velocity  of  any  part  of  a  linked  mechanism,  a  velocity  diagram 

may  be  drawn  as  illustrated  in  the  following  example.     AOB 

represents   diagi-ammatically    (fig.    113)    the    crosshead    of   a 

screw-steering  gear,  AC,  bd,  the  connecting  links,  and  c  and 

D  are  forced  by  guides  to  follow  the  axis  of  tho  frame  00. 

If  the  velocity  of  a  is  knov/n,  that  of  c  (or  any  other  part) 

can  be  found  ;  and  conversely. 


82 


ANGULAR   VELOCITY. 


Draw  oa  to  represent  the  vsloclty  of  A,  oa  being  perpen- 
dicular to  OA.  oh  in  the  oppoaite  direction  represents  that 
of  B.  The  velocity  of  c  relative  to  a  is  necessarily  perpen- 
dicular to  AC,  while  relative  to  the  frame  it  is  parallel  to  the 
axis.  Therefore,  draw  ca  perpendicular  to  ca,  and  oc  parallel 
to  the  axis  ;  this  gives  c.  Similarly  the  point  d  is  obtained. 
00  and  od  are  the  velocities  of  the  points  c  and  D.  The 
velocity  of  any  other  point,  say  E  in  the  connecting  link  AC, 
is  obtained  by  dividing'  ao  at  e  so  that  ae  :  ec  =  ae  :  EC. 
Join  oe,  which  is  the  velocity  of  the  point  E. 


Fig.  113. 


Fig.  114. 


\. 


'^^' 


If  /  be  the  middle  point  of  cd,  of  is  the  mean  velocity  of 
c  and  D,  i.e.  the  velocity  of  the  screw  shaft  as  a  whole,  to 
allow  for  which  a  small  amount  of  play  has  to  be  given. 
Note  that  the  shaft  is  moving-  towards  the  crosshead,  and 
that  the  velocities  of  C  and  d  relative  to  the  shaft  are  given 
by  jc,  df. 

Angular  Velocity. 

The  angular  velocity  of  a  body  about  an  axis  is  the  angle 
turned  through  about  the  axis  in  an  infinitesimal  space  of  time 
divided  by  that  time.  It  is  usually  expressed  in  radians 
per  second  or  in  rievolutions  p3r  minute,  the  unit  in  the  former 

fin 
case  being  —  or  9-55  times  that  in  the  latter. 

Composition  and  resolution  of  angular  velocities. — The 
angular  velocity  about  an  axis  may  be  represented  by  a  line 


ACCELERATION.  83 

drawn  parallel  to  the  axis,  and  of  Icng'tli  proportional  to 
the  magnitude  of  the  angular  velocity.  The  direction  of  the 
lino  usually  bears  the  same  relation  to  the  direction  of  rotation 
as  that  existing  between  the  travel  and  rotation  of  a  right- 
handed  screw.  Wlien  so  represented,  angular  velocities  are 
combined  and  resolved  in  the  same  way  as  linear  velocitiea 
(see  p.  81). 

Acceleration. 

The  acceleration  of  a  body  is  the  rate  of  change  of  its  velocity 
or  the  change  of  velocity  in  an  infinitesimal  space  of  time  divided 
by  that  time.  The  velocity  of  a  body  comprises  both  its  speed 
and  its  direction  ;  hence  the  acceleration  may  generally  be 
divided  into  two  parts — {a)  that  due  to  increase  of  speed,  which  is 

represented  by  -j-  ,  where  v  is  the  speed,  and  is  tangential  to  the 
dt 

direction  of  motion  ;  {b)  that  due  to  alteration  of  direction,  which 

is  directed  normally  towards  the  centre  of  curvature  and  is  equal 

to  v'^Ip  where  p  is  the  radius  of  curvature  of  the  path. 

Angular    acceleration  is  the    rate    of    increase    of    angular 

velocity;    and,    for    a    body  revolving  about  a  fixed  axis,   is 

represented  by  -^  where  u  is  the  angular  velocity. 
ut 

Composition  of  accelerations.  —  Accelerations,  linear  and 
angular,  are  combined  and  resolved  similarly  to  velocities. 

Acceleration  diagram  for  a  linked  mechanism.— To  find  the 
acceleration  of  any  part  of  a  linked  mechanism,  a  velocity 
diagram  is  first  constructed  as  in  fig.  113,  p.  82.  To  find  the 
accelerations  of  the  steering  gear  shown,  that  of  a,  assuming  the 

crosshead  to  revolve  uniformly,  is  equal  to—  or  —  ,  and  may  be 

represented  by  o'a'  parallel  to  AO.  The  acceleration  o'b'  of  OB  is 
equal  and  opposite.     The  normal  acceleration  of  c  relative  to  A  is 

represented  by  a'c^  equal  to  —  and  drawn  parallel  to  CA ;  the 

AC 

tangential  acceleration  of  c  relative  to  A  is  represented  by  c'-c' 
drawn  peipendicular  to  c^a' ,  the  length  of  c\'  being  at  present 
unknown.  The  acceleration  of  c  relative  to  the  frame  00  is 
parallel  to  the  axis  ;  so  that  o'c'  is  drawn  parallel  to  the  axis 
meeting  ch'  at  c',  giving  the  length  of  o'c'.  Similarly  the 
acceleration  of  D  is  found  by  drawing  b'd^,  and  d^d' ,  giving  o'd'. 
That  of  any  point  E  in  the  link  AC  is  obtained  by  joining  a'c'  and 
dividing  it  at  e',  so  that  a'e  :  e'c'  =  AE  :  EC.  Join  o'e',  which  is 
the  acceleration  of  the  point  E.  The  accelerations  of  C  and  D 
relative  to  the  screw  are  equal  to  ^  c'd'. 


84  DYNAMICS. 

DYNAMICS. 

Relations  between  Force  and  Motion. 

ox,  Oy,  02  =  Z  perpendicular  axes. 

P,  Q,  K  =  component  forces  along  ox,  oy,  Oz  acting  on  a  body. 

u,  V,  w  =  component  velocities  along  Ox,  Oy,  Oz. 

m  =  mass  of  body. 

M^,  My,   M2  =  momenta  parallel  to   ox,    oy,  oz  respectively  = 

inu,  niv,  mw. 
f,  g,  h  =  component  accelerations  parallel  to  Ox,  Oy,  oz. 
g  =  acceleration  due  to  gravity  =  32-2  in  foot-second  units. 

.         du      d^x         d?x    ^  dv      d^y        d^y 

■'  dt       dt  dt'  ^         dt       dt  dr 

,  dw       dMg  ^z 

IJote. — In  the  above,  if  m.  is  in  pounds,  p,  Q,  e  are  in 

poundals,  one  poundal  being  equal  -  pound  or  about  half  an 

ounce  weight  ;  if,  on  the  other  hand,  the  forces  are  expressed 
in  pounds,  the  mass  m  must  be  expressed  in  terms  of  the 
gravitational  unit  equal  to  g  (about  32)  pounds. 

Exam^ple. — A  force  of  2  lb.  acts  upon  a  masB  of  J31b.  ^  To 
find  the  acceleration. 

The  mass  in  gravitational  units  =  — 

P        2         2<7 

.'.  Acceleration' =  —  =  -7-  =  -f  =21|  ft.  per  second  per  second. 
-      in      2>\g       6  ^ 

Angulae  Motion. 
I  =  mass  moment  of  inertia  about  axis  of  revolution, 
i  =  angular  acceleration. 
«  =  angular  velocity. 
0  =  angle  turned  through. 
M  =  angular  momentum  =  l«. 
G  =  moments  of  forces  about  axis. 
_    y  _    doo        d-d      dM 

I^^ote. — If  G  is  expressed  in  foot-pounds,  I  must  be  expressed 
in  the  gravitational  unit,  or  is  i/gr  of  the  density  of  the 
material  multiplied  by  the  volume  moment  of  inertia  of  the 
body  (see  pp.  69-75). 

WOEK    AND    EnEEGY. 

The  work  done  by  a  force  on  a  body  is  the  product  of  the 
force  by  the  distance  moved  resolved  along  the  direction  of  the 
force.    The  work  done  by  a  couple  is  the  moment  of  the  couple 


DYNAMICS.  85 

multiplied  by  the  anjjle  turned  through  resolved  along  the  plane 
of  the  couple.  With  the  previous  notation,  if  the  body  runs 
through  distances  x,  y,  z  parallel  to  the  axes,  and  rotates  through 
an  angle  0,  the  work  done  is  vx -{- Q,y  -\- ixz  +  g9,  allowing  for  sign. 

The  energy  of  a  body  is  its  capacity  for  doing  work. 

Kinetic  energy  is  energy  due  to  motion.     With  the  preceding 

notation,  its  amount  is  ^  m  {u^+v'+td^)  +  5-  i»^  wi  and  I  being 

expressed  in  pounds  and  the  result  in  foot-pounds. 

Potential  energy  is  energy  due  to  position,  and  is  measured 
from  an  arbitrarily  fixed  datum.  A  body  of  height  h  feet  above 
the  sea-level  has  potential  energy  of  7iih  foot-pounds.  A  ship 
has  potential  energy  due  both  to  the  height  of  its  centre  of  gravity 
and  the  depth  of  its  centre  of  buoyancy. 

Molecular  energy,  due  to  heat,  electrical  state,  magnetism, 
vibration,  etc.,  is  frequently  waste  energy  as  far  as  its  capacity  of 
doing  useful  work  is  concerned. 

Conservation  of  energy.  The  work  done  on  a  bodv  (other  than 
that  involved  in  a  change  in  the  potential  energy)  in  a  given 
interval  of  time  is  equal  to  the  increase  of  its  total  energy. 

Power  is  the  rate  of  doing  work.  It  is  equal  to  Pw  +  Qy  + 
WW  -f  Gw,  allowing  for  sign.  This  is  equal  to  the  rate  of  increase 
of  energy.  The  practical  unit  of  power  is  the  hojse-poiver, 
equivalent  to  550  foot-pounds  per  second,  or  33,000  foot-pounds 
per  minute.  Another  unit  is  the  watt,  746  of  which  are  equivalent 
to  one  H.P. 

Uniform  Foece   in  Line   of  Motion. 
p  =  uniform  force  in  pounds  weight. 
ni  =  mass  in  pounds. 
/  =  uniform  acceleration  =  pgr/m. 

V  =  initial  velocity  in  feet  per  second. 
s  =  distance  travelled. 

t  =  time  occupied. 

V  =  final  velocity. 

v  =  Y+ft;  v''  =  \^  +  2fs',  s  =  \t  +  W. 

For  retarded  motion  change  /to  -/. 

For  motion  vertically  under  gravity  f  to  g  or  -  g,  according  as 

the  initial  motion  is  downwards  or   upwards.      In  that 
^  case  p  =  +m. 
For  'motion  down  an  incline  of  angle   o  to  the  horizontal, 

replace  f  hj  g  tan  o. 
For  angular  rotation  with  the  notation  above,   fl  being  the 

initial    angular    velocity,     u  =  n  +  ^t  \     a"  =  Cl}  +  2^9  ; 

e  =  m  +  ii^^ 

Gravity. 
g  —  acceleration  due  to  gravity  in  feet/second^. 
A  =  latitude  of  the  place. 
h  =  height  above  sea-level. 


86  DYNAMICS. 

R  =  radius  of  earth  in  feet  =  20,900,000. 


27i 


g  =  B2  088   (1  +  -005302    sin^  (p  -  -000007   sin''  2</>  -  —  ). 

R 

Usually  g  is  taken  as  32-2,  or  981  in  centimetres/second^. 


Simple  Vibration. 

M  =  mass  in  pounds. 

a  =  semi-amplitude  of  vibration. 

n  ■=■  frequency  or  number  of  double  vibrations  per  second. 

E  =  modulus  or  force  in  pounds   required  to  produce  unit 

extension. 
t  —  time. 

X  —  displacement  at  time  '  t.' 
/  =  acceleration  at  time  '  t.* 
a  =  a  constant. 
17  =  32-2. 


%i=i  ~\J  -^\  a;  =  a  sin  [lirnt  +  a) ;  / 


M 


Simple  Pendulum. 

L  =  length  of  pendulum  in  feet. 

T  =  time  of  a  single  small  vibration  in  seconds. 

g  —  acceleration  due  to  gravity  =  32-2. 

T  =  7r\/-=  -554  v/£: 


Table  giving 

THE  Lengths  of  Pendulums  in  Inches    1 

THAT  Vibrate  Seconds 

IN  various  Latitudes. 

Sierra  Leone 

89-01997 

New  York 

39-10120 

Trinidad 

39-01888 

Bordeaux 

3911296 

Madras 

39-02630 

Paris 

39-12877 

Jamaica 

39-03o03 

London 

3913907 

Eio  Janeiro 

39-04350 

Edinburgh 

39-15504 

Table  giving  the  Times  of  Vibration  for  Pendulum 
swinging  through  Large  Arcs. 

Angle  swung  on  each 
side  of  vertical 

80" 

CO' 

90P 

120° 

150" 

180' 

Actual  time  of 
vibration  -r  Time  for 
infinitely  small  angle 

1017 

1073    1-183 

1-373 

1-762 

Infinite 

DYNAMICS.  87 

Compound  Pendulum. 
K  =  radius  of  gyration  of  body  about  axis  of  rotation. 
h  =  centre  of  gravity  below  axis. 
I  =  length  of  equivalent  pendulum. 

I  =  k-jh. 
The  centre  of  'percussion,  or  point  at  which  a  blow  struck 
perpendicularly  to  the  axis  will  cause  no  stress  at  the  axis,  is 
situated  at  a  distance  I  (determined  by  the  above  formula) 
below  the  axis. 

Centrifugal  Force. 
F  =  centrifugal  force  of  body  revolving  in  a  circle  at  a  unifcrra 
speed,   or  apparent  force  required  to  balance  that  necessary  to 
produce  the  requisite  normal  acceleration. 
w  =  weight  of  body. 
N  =  number  of  revolutions  per  minute. 
n  =  number  of  revolutions  per  second. 
V  =  linear  velocity  in  feet  per  second. 
«  =  angular  velocity  in  circular  measure  per  Kecood. 
r  =  radius  of  circle  in  feet. 
g  =  acceleration  due  to  gravity  =  32-2  nearly. 
_  wu^  _  viraP  _  4wnVV  _  v.'nV  _  wnV 
~  ~gr   ~     g      ~        'g        ~  ^8154"  2935 

Gyroscopic   Action. 

If  the  axis  of  a  revolving  body  is  made  to  rotate  into  a  new 
position,  resistance  is  experienced  due  to  the  '  gyroscopic 
action  '  of  the  revolving  mass.  Let  ab  represent  in  the  usual 
way  the  angular  momentum  ia>  of  a  body  having  a  moment 
of  inertia  i  about  the  axis  of  revolution,  and  an  angular 
velocity  u.  If  this  axis  is  forced  to  occupy  after  a  short  time 
the  position  AC,  BC  represents  the  chajige  of  angular 
momentum.  This  is  e^qual  to  iw  x  Lb  AC.  If  this  change  is 
effected  by  turning  the  axis  uniformly  with  angular  velocity 
w',  the  rate  of  change  of  angular  momentum  is  Icow',  which  is 
equal  to  the  moment  G  of  the  applied  couple.  Note  that  the 
plajio  of  G  is  perpendicular  to  that  of  shaft  rotation,  and  of 
the  direction  of  movement.  If  i  is  in  weight  units  (lbs.  X 
feet^),  and  n  and  n'  are  the  number  of  revolutions  per  minute 
of  shaft  rotation,  and  of  bodily  rotation, 
_  I  47r^  nn'  _  i.n.n' 
^  ~'g^    3600     "   2935 

In  the  case  of  a  ship  going  ahead  with  a  right-handed 
screw,  the  forces  required  on  the  shaft  when  turning  to 
starboard  are  downward  aft  and  upward  forward  ;  the  re- 
action on  the  hull  is  then  such  as  to  cause  a  slight  trim  by 
the  bow. 


88 


HYDROSTATICS. 


Wl 


Impact. 
M2  =  the  velocities  of  two  bodies  before  impact  (if  moving 
in  opposite  directions  make  U2  negative). 
Vi,  1)2  =  the  velocities  after  impact. 
mi,  ni2=  the  masses. 

e  =  coefficient  of  restitution  =  ratio  of  velocity  of  separa- 
tion to  that  of  approach. 
For  direct  impact, 

_  ui  (mi  -  g^na)  -f  ma  M2  (1  +  e) 


Vi 


V2 


mi  +  ma 
Ui  mi  (1  +  e)  +  U2  (wia 


enii) 


Kinetic  energy  lost  = 


mi  +  TTta 
—  ^^1  ^^  (^^1  -  ih)"  (1  -  e^) 
2g  (mi  +  m^2) 

Total  momentum  is  unchanged,  or  7ni  ui  +  ma«2  —  ^'^i  Ui  +  w^2  v^. 
For  oblique  impact,  resolve  the  velocities  along  and  perpen- 
dicular to  the  line  of  impact  ;  treat  the  components  along 
the  line  by  the  above  formulae  ;  the  latter  are  unaltered  by 
the  impact. 

The  value  of  the  coefficient  e  depends  to  some  extent  on 
the  shape  of  the  bodies  and  the  velocity  of  impp,ct,  as  well 
as  on  the  material.  Approximate  values  for  the  impact  of 
like  materials   are   given   in   the   following   table  :  — 


Material 

Cast  Iron 

Mild  Steel 

SoffBrasa 

Lead 

Elm 

Glass 

Ivory 

e 

•70 

•67 

•38 

•20 

•60 

•94 

•81 

HYDROSTATICS. 

The  density/  of  a  fluid  is  the  wfeight  of  a  unit  volume., 
Generally  it  is  stated  in  pounds  per  cubic  foot,  or  inversely 
aa  the  number  of  cubic  feet  required  to  weigh  1  ton.  (See 
tables  on  p.  262.) 

The  specific  gravity  of  a  fluid  is  the  ratio  of  its  density  to 
that  of  water. 

Density   of   a   Mixture   of   Two   Liquids. 
Wi,  W2  =  densities  of  the  two  liquids. 

w  =  density  of  the  mixture. 
mi,  m2  =  proportion   of   the  two  liquids   in  the 

volume, 
ni,    na  =  proportion  of  the   two   liquids   in   the 
weight. 

mi  wi  +  7)12102  _  ni_ 
mi  +  r?ta  ni 

Wi 


ni/wa  =  mi Wilm2  W2',  w  = 


mixture  by 

mixture  by 

+  Ha 
na 

W2 


HYDROSTATICS.  89 

Peessure  in  a  Liquid. 

w  =  density  of  liquid  in  pounds  per  cubic  foot. 
z  =  depth  below  free  surface. 

p  =  intensity  of  pressure  in  pounds  per  square  inch. 
P  =  intensity  of  pressure  in  pounds  per  square  foot. 
V  =  wz ;  p  =  wzjlii. 

4 
In  salt  water,  w  =  Gi,  v  =^  Giz,  p  =  ^z. 

In  fresh  water,  w  =  62-5,  p  =  62-5^,  p  =   iSSz. 

If  the  absolute  pressure  bo  required  P  and  p  must  be 
increased  by  2120  and  14'7  respectively,  in  order  to  allow  for 
the  pressure  of  the  atmosphere. 

Nofe.-^The  centre  of  pressure  of  an  immersed  plane 
surface  is  that  point  on  the  surface  through  which  the 
resultant  pressure  acts. 

Pressure  on  Immersed  Plane  Surface. 

If  surface  be  vertical  find  the  centre  of  gravity  a  and 
take  axes  Gx  horizontal  and  Gy  vertically  downwards.     Let 
A  =  area  of  plane. 

h  =  depth  of  centre  of  gravity  belowfree  surface. 
w  =  density  of  fluid. 
T  =  total  thrust  or  pressure  on  plane. 
X,  y  =  co-ordinates  of  the  centre  of  pressure. 
Then  t  =  wA.h 


y  -  Ah 
1 


~j  /  y^  .  dx  .  dy  over  area. 


,   X  moment  of  inertia  of  area  about  Gx. 
Ah 


X  =  — r  /  a:?/  dx  dy  over  area. 


=  -r  X  product  of  inertia  of  area  about  Gx,  Gy. 
Ah 

If  the  surface  and  the  axis  Gy  be  inclined  at  an  angle  9  to  tl;e 
vertical,  T  and  x  are  unaltered,  but  the  value  found  for  y  should 
be  multiplied  by  cos  0. 

Pressure  on  any  Closed  Surface. 

The  resultant  pressure  on  the  whole  immersed  surface 
of  a  body  Is  equal  to  the  weight  of  the  water  displaced  by 
the  body  and  acts  vertically  upwards  through  the  centre  of 
gravity  of  the  displaced  volume.  The  upward  force  is  termed 
the  displacement,  and  the  point  through  which  it  acts  the 
centre  of  buoyancy. 


90  DISPLACEMENT, 

DISPLACEMENT,  Etc. 
Computation  of  a  Ship's   Displackmext. 

This  consists  of  computing  the  volume  of  the  body  of  the 
vessel  below  the  water-plane,  up  to  which  it  is  required  to 
know  her  displacement,  by  one  of  the  rules  used  for  finding 
the  volume  of  solids  bounded  on  one  side  by  a  curved  surface 
(see  pp.  54,  55). 

Two  processe3  are  generally  made  use  of  in  computing 
a  vessel's  displacement,  as  the  calculations  in  each  process 
are  required  to  determine  the  position  of  the  centre  of  gravity 
of  displacement,  or  centre  of  buoyancy,  and  also  because  thp 
two  results  (are  a  check  on  the  correctness  of  the  calculations. 

One  process  consists  in  dividing  the  length  of  the  ship  on 
the  ^  load  water-line  by  a  number  of  equidistant  vertical 
sections,  computing  their  several  areas  by  one  of  Simpson's 
rules,  and  then  treating  them  as  if  they  were  tlie  ordinates 
of  a  new  curve,  the  base  of  which  is  the'  load  water-line. 

The  other  process  consists  in  dividing  the  depth  of  the 
vessel  below  the  load  water-line  by  a  number  of  equidistant 
horizontal  planes  parallel  to  the  load  water-line  ;  the 
areas  of  their  several  planes  are  then  computed  by  one  of 
Simpson's  rules,  knd  those  areas  are  treated  as  if  they  were 
the  ordinates  of  a  new  curve,  the  base  of  which  is  the  vertical 
distance  between  the  load  water-line  and  lowest  horizontal 
plane. 

As  the  vessel  generally  consists  of  two  symmetrical  halves, 
the  volume  of  only  half  the  vessel,  below  the  load  water-line, 
is  calculated,  the  ordinates  all  being  measured  from  a  longi- 
tudinal vertical  plane  at  the  middle  of  the  ship. 

Usually  the  portion  below  the  lowest  water-line  is  treated, 
as  are  also  the  stern,  rudder,  bilgea,  keels,  oto.,  as  an 
appendage,  its  volume  being  calculated  by  means  of  equi- 
distant vertical  sections.  The  water-lines  that  are  '  snubbed.  * 
or  cut  short  abaft  the  fore  perpendicular  or  before  the  after 
perpendicular  are  conceived  to  extend  to  these  perpendiculars, 
the  extra  volumes  thus  introduced  being  regarded  as  negative 
appendages. 

The  displacement  of  a  ship  can  also  be  obtained  by  dividing 
the  length  into  sections,  spaced  as  required  by  Tchebycheff's 
rule  ;  the  integration  in  a  longitudinal  direction  is  effected 
by  simple  summation.  The  water-lines  are  equidistantly  spaced 
and  integrated  by  Simpson's  rules  as  before.  This  method  is^ 
generally  speaking,  more  expeditious  than  is  the  one  pre- 
viously described,  since  fewer  ordinates  can  be  employed,  and 
half  the  multiplication  is  dispensed  with. 

Both  methods  are  illustrated  in  the  displacement  Sheet? 
given  on  pp.  94  ff. 


DISPLACEMENT. 


91 


Determination  of  a  Ship's  Centre  of  Buoyancy  for 
THE  Upright  Position. 

The  centre  of  buoyancy  is  also  termed  the  centre  of  gravity 
of  displacement,  as  it  occupies  the  same  point  as  the  centre  of 
g^ravity  of  the  volume  of  water  displaced  by  the  vessel,  and  its 
position  is  determined  by  the  rules  used  for  finding  the  centre 
of  gravity  of  solids,  bounded  on  one  side  by  a  curved  surface 
(see  rules,  pp.  66  and  67),  with  the  exception  that  its  position  need 
only  be  determined  for  its  vertical  distance  from  a  horizontal 
plane,  and  its  horizontal  distance  from  a  vertical  plane  ;  for  the 
ship  consisting  of  two  symmetrical  halves,  it  must  necessarily 
lay  in  the  longitudinal  vertical  plane  in  the  middle  of  the  ship. 

Calculation  of  the  centre  of  buoyancy  is  generally  performed 
on  the  displacement  sheet  (see  pp.  94  ff.). 


Curve  of  Areas  of  Sections. 

This  curve  (see  fig.  115)  is  of  use  in  designing  and  in 
estimating  the  resistance  of  a  ship,  for  it  fixes  the  distribution 
of  displacement  along  the  length. 

Fig.  115. 


CURVE  OF  SECTIONALAREAS. 

Method  of  Construction. — Compute  the  area  of  each 
transverse  section  up  to  the  l.w.l.;  and  set  it  oif  to  scale 
on  a  base  of  length.  A  curve  drawn  through  the  tops  of  the 
ordinates  will  form   the  curve  required. 

Curve  of  Areas  of  Midship  Section. 

This  curve  (see  fig.  116)  is  used  to  determine  the  area  of  the 
immersed  part  of  the  midship  section  of  a  vessel  at  any  given 
draught  of  water. 

Method  of  Construction. —  Com\mXe  the  areas  of  the  midship 
section  from  the  keel  up  to  the  several  longitudinal  water-planes 


Fig.  116. 


KrtHe  •/  areat 


which  are  used  for  calculating 
the  displacement  ;  set  these 
areas  off  along  a  base  line  as 
ordinates,  in  their  consecutive 
order,  the  abscissa?  of  which  re- 
present to  scale  the  respective 
distances  between  the  longi- 
tudinal water-planes:  a  curve 
bent  through  the  extremities 
MLjs"  of  these  ordinates  will  form 
the  required  curve. 


92 


CURVE   OF    DISPLACEMENT. 


Curve  of  Displacement. 

This  curve  is  used  to  determine  the  displacement  a  vessel 
has  at  any  draught  of  water  parallel  to  the  load  water-line 
(see  fig  117). 

Method  of  Construction. — This  curve  is  constructed  in  a  similar 
manner  to  the  foregoing  curve,  with  the  exception  that  the  ordi- 

FiG.  117 


tcale  of  Tons 


nates  represent  the  several  volumes  of  displacement  (in  tons  of 
35  cubic  feet  for  salt  water,  and  36  cubic  feet  for  fresh  water) 
up  to  their  respective  longitudinal  water-planes. 

Curve  of  Tons  ter  Inch  of  Immersion. 

This  curve  (see  fig.  1]  8)  is  used  to  determine  the  number  of 
tons  required  to  immerse  a  vessel  one  inch  at  any  draught  of 
water  parallel  to  the  load  water-plane. 

To  find  the  displacement  per  inch  in  cubic  feet  at  any  water- 
plane,  divide  the  area  of  that  plane  by  12 ;  and  if  the  displace- 

FiG.  118. 


Scale  of  Tons 


ment  per  inch  is  required  in  tons,  divide  by  35  or  36,  as  the 
case  may  be. 

A  =  area  of  longitudinal  water-plane  in  square  feet. 

T  =  tons  per  inch  of  immersion  at  that  water-plane. 

T  =  —~ —  for  salt  water  ;  T  = , for  fresh  water. 

115x35  '12x36 


COEFFICIENTS   OF   FINENESS. 


93 


Bthod  of  C (instruction. — This  curve  is  also  constructed 
ixi  a  similar  manner  to  the  two  foregoing  curves,  wifch  the 
exception  that  tlie  ordinafces  represent  to  scale  the  tons  per 
inch  of  immersion  at  the  respective  water-pianos. 

The  coefficients  of  fineness  of  a  vessel  consist  of  the  block 
coefficients  {&),  the  prismatic  coefficient  (7),  and  the  midship 
section  coefficient  (/t).  They  are  determined  from  the  following 
equations  : — 

V  =  volume  of  displacement  in  cubic  feet. 

L  =  length   of  vessel  at  load   water-line  in  feet  (or  length 

between  perpendiculars,  according  to  convention). 
B  =  extreme  immersed  breadth  in  feet.     (Occasionally  this  is 
taken  as  the  breadth  at  l.w.l,  in  cases  where  this  is 
less  than  the  extreme  breadth.) 
D  =  mean  draught  of  water  in  feet.     (Take  to  top  of  keel  if 

bar  keel.) 
2  =  Area  of  midship  section  up  to  l.w.l.  in  square  feet. 
o  V  V  2        ^ 

^  =  L":B:^.'"''"i:2'^=i:^.'^  =  '>'-'*- 

Another  coefficient  sometimes  used  is  that  of  water-line  area 
(a)  given  by  A  =  :^— ^  where  A  is  the  area  of  the  l.w.l.     Usually 

L.B. 


tons  per  inch' 


this  latter  being 


this  is  expressed  as  a  coefficient 
equal  to  -— - 

A 

Values  of  these  four  coefficients  for  typical  ships  are  given 
in  the  table  below. 


Table  01 

'  Coefficients  of  Fineness. 

Block  Co- 

Prismatic 

Mid.  Sec. 

Waterline 

efficient 

«        V 

Coefflc't 

V 

Coefflc't 
2 

Coefflc't 

^       A 

Class  of  Ship 

Tons 

^-I.BD 

"^=£2 

^-^ 

^=Zi 

per  Inch 

Battleship  (modern)     . 

•GO 

•62 

•965 

•73 

575 

Battleship  (older) 

•65 

•68 

.95 

•81 

620 

First-class  Cruiser 

•56 

•62 

•90 

•68 

620 

Modern  Light  Cruiser  . 

•58 

•63 

•92 

•76 

550 

T  -ncdo  Boat  Destroyer 

•55 

•67 

•82 

•76 

550 

a  Yacht 

•52 

•565 

•92 

•69 

610 

:  .   •  Passenger  Steamer 

•59 

•62 

•95 

•70 

600 

Large  Cargo  Vessels     . 

•73 

•77 

•95 

•83 

510 

Sailing  Yacht 

•2 

•5 

•4 

•75 

660 

Tug     . 

•58 

•61 

•95 

•76 

550 

A^o^«.— The  '  length  '  in  warships  is  the  length  between  perpendiculars. 


Table   showing   Method   of  Computing  a   Ship 


Length  between  perpendiculars,  885  feet 

;  breadth.  41  feet ;  draught  at  perpendicula 

1 

Appendage  below  lowest  Water-line 

Wateb-lines 

1 

3 

■< 

li 

i!i 

1-2 

it 

7W.L. 

6W.L.            6W.L.            4W.L. 

3W.L 

3 

Simpson's  Multipliebs 

3 
c 

i 

2 

1 

2 

1 

h 

•9 
2  05 
3-67 
5-24 
7-18 
9-88 
12-48 
14-3 
15^ 
15-7 
14-2 
11-35 
8-1 
5-72 

•9 
4-1 
3-67 
10-48 
7-18 
19-36 
12^ 
28-6 
15-65 
31-4 
14-2 
22-7 
8-1 
11-44 
3-9 
5-25 
•27 

10 
~9 
8 
"7 
~G 
"5 
~4 
"3 
"2 
1 

"o 

1 
1 
3 
4 
~5 
~6 
"7 
1 
~9 

1-6 

•8 

1-6 
6-0 

9^2 
12-8 

8-46 
•20-6 
11-96 
26-2 

1^76  3^52 
3^52 

•8 

•8 
2-6 
2-6 
4-66 
4-66 
6-94 
6-94 
9-4 
9-4 
11-7 
11-7 
13-9 
13-9 
15-86 
15-86 
17-34 
17-34 
18-26 
18-26 
18-7 
18-7 
18-74 
18-74 
18-06 
18-06 
17-1 
17-1 
15-4 
15-4 
13-2 
13-2 
10-3 
10-3 
6-9 
6-9 
3-8 
3-8 
1-5 
1-5 
•76 
•76 

•4 

52 

4-66 
13-88 

9-4 
23-4 
13-9 
31-72 
17-34 
33^52 
18^7 
37-48 
18-06 
34-2 
15-4 
26-4 
10-3 
13-8 

3-8 

3-0 
•38 

•8 
16 
30 

6^0 
5-3 
10^6 
79 
15^8 
10-5 
21-0 
12-86 
25-72 
15-04 
30-08 
16-9 
33  •  8 
184 
36^8 
193 
38^6 
197 
39^4 
1964 
:39-28 

•4 

60 
5-3 
15-8 
10-5 
25-72 
15-04 
33-8 
18-4 
38-6 
19-7 
39-29 
19^2 
36^92 
17-0 

•5 

•5 

31 

3-1 

5-7 

5-7 

8-54 

8-54 

11-2 

11-2 

13-66 

13-66 

15-8 

15-8 

17-6 

17-6 

190 

19-0 

199 

19-9 

20-26 

20-26 

20-24 

20-24 

19-9 

19^9 

193 

193 

181 

181 

2 

— 

6 

1 

7-2 

•3 

•27 

3-6 

7^0 
55 
11^0 

j  3-5 

1 

11^0 

1 

5 

2 

28-7 

-45 

1^84 

3-0 

1-5 

4-6 

2-3 

6-4 

3-2 

8-46 

4-23 

10-3 
5-15 

11-96 
5-98 

13-1 
6-55 

17 

1 

22-02 

•48 

1^76 

7-64   7-64 
15^28 

11 

2 

52-4 

•51 

5^34 

9-9 

19^8 

|19^8 

27 

1 

28-72 

•54 

3-87 

12^04  12^04 
24^08 
141  ,282 
28-2  1 
15-6  jl5-6 
31-2  1 

15 

2 

59-28-57 

11^25 

35 

1 

24 -9o 

•6 

7^5 

19 

2 

28-6 

-63 

18^0 

1654 

33-08 

33-08 

17-1 

340 

16^2 

29-6 

12-8 

20^8 

7-3 

8-68 

2-1 

1^6 
•38 

39 

1 

251-88 

•66 

10-33 
20-7 
9-22 
14-8 

13-6 
6-8 

13-46 
6-73 

12-5 
6-25 

10-6 
5^3 
8-3 
4-15 
5-9 
2-95 

136 
26^92 
12^5 
•21  •  2 

8^3 
11^8 

3^7 

17-1 
34-2 
17-0 
34-0 
16-2 
32-4 
14-8 
29-6 
12-8 
25-6 
10-4 
20-8 
7-3 
14-6 
4-34 
8-68 
2-1 
4-2 
-8 
1-6 
•76 
1^52 

20 

2 

31-4 

-66 

40 

1 

28-4 

-65 

19-2 

38-4 

18-46 

36-92 

170 

34^0 

151 

30^2 

12-64 

25-28 

9-5 
19-0 

6-0 
12-0 

30 

6-0 
-76 

1-59 

19 

2 

68-1 

-67 

38 

1 

32-4 

5-58 

18 

2 

57-2 

-71 

8-14 

30-2 

164 

16^4 

32 

1 

3-9 

2-63 

•54 

23-4 

-73 

2-88 

3-7 

1^85 

12-6414-34 

19-0    11-6 
11-6 

6-0  1  8-4 
i  8-4 

60  1  5-0 
1  5-0 

14 

2 

36-82 
2-16 

•75 

3-94 
•081 

22 

1^1 

4^4 
1^0 
1^52 

•38 

23 

I 

1-0 

•5 
•76 

•38 

8 

2 

— 

— 

10 

1 

— 

— 

10     — 

— 

— 

•76 

-38 

•38 

15 

1^5 

1 

200^0  279^8S 
251^88 
2800 


125-47    197-54    284-94    337-94    375-88    404 

98^77  +  569-83  +  337-94  +  751-76  +  404 

6   5   4   3   

592^62  -f  2849-40  +  1351-76  +  2255-28  +    808 

>}.B.— The  dark  figures  are  the  ordinates;  the  light  figures  under  them  and  also  to  their  right  are  t] 
)duct8  of  the  ordinates  by  their  respective  Simpson's  multipliers,  which  are  ilaced  at  the  head  ai 
the  left  of  the  table ;   if  each  row  and  column  of  these  products  be  added  together,  and  the  resul 


Displacement,   etc.,   using   Simpson's   First  Rule. 

B  feet  for'd,  14  feet  aft,  13  ft.  6  in.  mean ;  waterlines  apart,  2  feet ;  ordinates  apart,  19  ft.  8  in. 


24 


46irr 
921  _ 
14: 20 

28!_ 
6610' 
12 


6-32 


66 


28 


26 


360 
19^ 
40^ 


20-46 


28-92 


6-6 


6-34 


18' 


145-71 


19-36 


40-4 


20- 


41-0 


19-04 


2416- 
12 


12-1 


18-0 


2-58 


Vertical  Sectioks 


3-3 
23-19 


44-01 


66-29 

88-47 


109-59 


158-6 


166-77 


170-53 
170-24 


165-64 


157-57 


126-83 


105-41 


80-71 


56-07 


33-18 


14-74 


S 

ft 

11 


_1 

46-38 


44-01 


132-58 


88-47 


218-18 
128-88 


-291-42 


158-6 


333-54 


170-53 


340-48 


165-64 


315-14 


144-17 


253-66 


105-41 


161-42 


56-07 


66-36 


7-37 


10 


16-5 


417-42 
.352-08 


928-06 


9-26 
11-94 
14-46 
515-52  16-54 
874-26  18-2 


530-82 


1095-9 


317-2 


333-54 


5381-3 


340-48 


381-28  20-3 
945-42  19-9 


576-68 


1268-3 


632-46116-24 
1129-94 


448-56 


597-24 


73-7 


Metacentkes 


19-36 
20-2 
20-5 
20-5 


19-04 
17-76 


14-3 
120^ 
-0 


•16 


1702 
3023 
4525 
6029 
7256 
8242 
8615 
8615 


7881 
6902 
5600 
4283 
2924 
1772 
729 


138 


6-6 
6-34 

1586118-52 
1702111 -94 
6046128-92 
16-54 


4525 


12058  36-4 


8365  20 


6902 


4283  16 


5848  28 


Longitudinal 


7256  19-36 
16484  40-4 

8615  20-5 
17230  41-0 


15762139-8 
19-04 
11200|35-52 


1772|L2-1 
1458118-0 
2-5 


S2I 

.2  o(i( 


59-4 
50-72 


129-64 


71-64 


144-60 


66-16 


109-2 


38-72 


40-4 


710-48 


41-0 


40-6 


119-4 


76-16 


177-6 


97-44 


200-2 
96-8 


162-0 


25-8 


2  ©""  n- 


10 

9 

534-6 

8 

405-7 

7 

907-4 

6 

429-8' 

5 

723-0 

J 
3 

264-64 
327-6 

2 

77-44 

1 

40-4 

0 

0 

1 

410 

2 

81-2 

3 

358-2 

4 

304-64 

5 

888-0 

6 

584-64 

7 

1401-4 

8 

774-4 

9 

1458-0 

258-0 


424-48  438-7  6344-06  131488  438-7    103700     9860-24 

848-96    +    219-35     =     3231-0  5381-30  710-48 

1    _  962-76  326-52 

848-96    -    8706-78  (Continued  on  next  page.) 

grated  by  the  proper  multipliers,  and  the  sums  of  these  products  added  tocether,  the  two  sums 
agree  if  the  calculations  are  correct.    In  tliis  case  the  sum  thus  obtained  by  two  methods  is  82310. 


96 


DISPLACEMENT. 


T^•     1  J.  .        ,.,      3231-0      8      ,„  „„ 

Displacement  -  mam  solid  =     gg      x  -  x  19-25  x  2  ==  3159  tons. 

8706 • 78     8 
Moment  belo\v  L.w.i..  -  main  solid  =      g^      x  -  x  19-25  x4  =  17,026ft.-tons. 

g 
Moment  abaft  3^  -  main  solid  =  962-76  x  -  x  (19-25)2  x  2  =  18,121  ft.-tons. 

200      4 
Displacement  —  lower  appendage  =  xr-  x  —x  19-25  =  147  tons. 

do  O 

125 • 47 
C.G.  below  7  "W.ii.  —  lov/er  appendage  =    g^    =  -63  feet. 

Moment  abaft  ^  -  lower  appendage  =  ^  ^  3  x  (19-25)2  =  395  ft.-tons. 


Below 

L.W.L. 

Abaft 

^ 

Item. 

Tons. 

Distance. 

Moment. 

Distance. 

Moment. 

Main  solid 

3159 

17026 

18121 

Lower  appendage 

147 

12-63 

1857 

395 

Aft 

5-7 

1-3 

7 

197-8 

1127 

Fore             „ 

-6 

6-0 

4 

-  193-6 

-  116 

Rudder 

2-2 

8-7 

19 

194-3 

427 

Bilge  keels 

•6 

9-5 

6 

Shafting 

3-2 

6-3 

20 

153-8 

492 

Shaft  brackets 

1-2 

6-5 

8 

168-5 

202 

Propellers 

•9 

6-7 

6 

173-1 

156 

gwell 

•6 

5-8 

3 

121-5 

73 

Recess 

-2-8 

5-9 

-16 

146-6 

-  410 

Negative  appendage 

a-ft 

-7-0 

11-2 

-78 

176-0 

-1232 

Total 

3311-2 

5-70 

18862 

5-8 

19235 

Displacement  3311  tons ;  c.B.  5-70'  below  li.w.li.,  5-8'  abaft  }£. 

4 
Area  of  l.w.l.  (main  portion)  =  438-7x-xi9.25  =  11,260  sq.  ft. 

Moment  abaft  ^  -  32G-52  x  |-  x  (19-25)2  =  161,300  ft.3 

Moment  of  inertia  about  ^  -  9860-24  x  ^  x  (19-25)3  =  93,700.000  it* 


Item. 

Area. 

c.F.  abaft  }^ 

Moment 
about  ^ 

Moment  of  Inertia 
about  )^ 

Main  portion 
Appendage  aft 

11,260 
100 

198 

161,300 
19,800 

93,700,000 
3,900,000 

Total 

11,360 

15-95 

181,100 

97,600,000 

11,360  X  (15 -95)2  =    2,90q,_p00 
Moment  of  inertia  about  c.F.  =  94,700,000  ft.'* 


DISPLACEMENT   SHEET. 


97 


11,350  K^ 

420~  ""  27  0.  c.F,  abaft  ^  =  15-95'. 

1.1  nrm  nnf\ 

BG  (with  G  in  li.w.L.)  =  6'  approx. 


Tons  per  inch 

T        •     ;,•     ,  94,700,000     „,  , 

Longitudinal   bm  =  „.,.-   ^-^  =  818 

diill  X  do 
.*.  Longitudinal  gm  =  818  -  6  =  812. 

■J311  X  81'' 
Moment  to  change  trim  1  inch  =  -.     ^  ^^  =  582  ft. -tons. 

Transverse  bm  =  |f^'J^  x  |  x  .|  x  19.25  =  9-78  ft. 
4 


Area  of  midship  section 

=  170-5  X- 

x2  = 

154  sq 

.ft. 

No. 

S.M 

'A.'\V.L.(2ft.aboveL.AV.L.) 

2W.L. 

:>  W.L. 

X 

IJ 

1 

5 

Li 

V  0  — 

1 

1 

■|o| 

1 

1 

|J 

1 

h 

1 

§  t 
fa  0 

1 " 

0 

0 

1  ^ 

1 

3 
0 

.— 

_ 

_ 

•2 

•5 

2 

2 

3-45 

6-90 

41 

82 

3-16 

31 

62 

31 

30 

60 

3 

1 

6-65 

6-65 

294 

294 

6-04 

220 

220 

5-7 

185 

185 

4 

2 

9-55 

19-10 

871 

1742 

8-94 

717 

1634 

8-54 

623 

1246 

5 

1 

12-2 

12-20 

1816 

1816 

11-66 

1581 

1581 

11-2 

1405 

1405 

6 

2 

14-65 

29-30 

3144 

G288 

14-14 

2833 

5666 

13-66 

2549 

5098 

7 

1 

16-7 

16-70 

4657 

4657 

16-26 

4291 

4291 

15-8 

3944 

3944 

8 

2 

18-25 

36-50 

6078 

12156 

18-0 

5832 

11664 

17-6 

5452 

10904 

9 

1 

19-3 

19-30 

7189 

7189 

19-3 

7189 

7189 

19-0 

6^59 

6859 

10 

2 

2.J-05 

40-10 

8060 

16120 

20-14 

8181 

16362 

19-9 

7881 

15762 

11 

1 

20-4 

';0-40 

8490 

8490 

20-46 

8552 

8552 

20-26 

8316 

8316 

12 

2 

20-4 

40-80 

8490 

16980 

20-5 

8615 

17230 

20-24 

8292 

16584 

13 

1 

20-2 

20-20 

8242 

8242 

20-24 

8300 

8300 

19-9 

7881 

7881 

14 

2 

19-85 

39-70 

7821 

15642 

19-7 

7645 

15290 

19-3 

7189 

14378 

15 

1 

19-1 

19-10 

6968 

6968 

18-7 

6539 

6539 

18-1 

59;i0 

5930 

16 

2 

18-0 

36-00 

5832 

11664 

17-2 

5088 

10176 

16-4 

4411 

8822 

17 

1 

16-65 

16-65 

4616 

4616 

15-46 

3690 

3690 

14-34 

2950 

2950 

18 

2 

14-95 

29-90 

3341 

6682 

13-14 

2274 

4548 

11-6 

1561 

3122 

19 

1 

13-05 

13-05 

22-22 

2222 

10-56 

1174 

1174 

8-4 

593 

593 

20 

2 

10-25i  20-50 

1(<77 

2154 

7-1 

358 

716 

5-0 

125 

250 

21 

§ 

6-6 

8-30 

287 

\44 

3-24 

34 

17 

1-5 

8 

2 

446-35 

1^4148 

124701 

114291 

Function  Mult.  Product.  Mult.  Moment, 
of  area. 


w.r. 

NT.L. 


Up 
yer  i., 


416-4 

4.38-7 
424-5 


2232 
3510 
5742 

424 

5318 


3125 
2632 
5757 
_424 
5333 


Displacement  of  layer 

(L.W.L.  to  A.W.L.)  = 

5318  x|x  19.25x^^x1 


to  L.W.L. 
W.Ii.  to  A.W.L. 


Tons. 

3311 

651 
S962 


Moment 
helotv  L.w.L, 


651  tons. 
Moment  about  l.w.l.  — 
5333x|xl9.25xixl  = 
652  ft.-tons. 
Displacement  to  A.W.L.  =  3962  tons. 
,    ,  1^210      ,  ^  , 

c.b.  below  L.w.L.  =  ^ofio'^*"^ 

4     19-25 
Tonsperin.=446-4x-x-^  =27-3. 

1.34148       4     19-25 
Transverse  BM^g-^^-^  ^  i  ^  ~F~  = 

8 -28'. 


98 


DISPLACEMENT  SHEET. 


2W.L. 


Function  Mult.  Product.  Mult.  Moment, 
of  area. 


L.W.I.         438-7 

5 

2194 

3           1316 

.'W.L.         424-5 

8 

3396 

10            4245 

3W.L.         404-4 

-1 

5590 

-1            5561 

4  4 

404 

5186 

5157 

Tons. 

Movient 
about  li.w.ii. 

Up  to  L.W.Ij. 

3311 

18862       Toi 

Layer  l.w.l.  to  2 

W.T,. 

634 

630      Trai 

2677 

18232 

Displacement  of  layer 
(l.w.l.  to2w.ry.)  = 

5186x|xi9-25xf2X^  = 

634  tons. 

Moment  about  li.w.L.  = 

5157  x|x  19-25x^^x1  = 

680  ft.-tons. 


Displacement  to  2  w.l.  =  2677  tons, 
18232 
c.B.  below  L.w.L.  =^^  =6*82'. 

4      1 Q  -  Q.*} 
Tons  per  in. =424 -5  X  -><  "T^TT  =25-9. 
3       420 
124701       4     19-25 
Transverse  BM=2g^-^-^  ^  3  "^  ""§"  = 
11-4'. 


J  W.L.. 

Ci.W.Ii. 
!W.L. 
JW.L. 


Functicn  Mult.  Product.  Mult.  Moment, 
of  area. 


43S-7 
424-5 
404-4 


438-7 
1698-0 

25n-i 


Displacement  of  layer 
(li-W.Ii.  to  3w.L,.)  = 

2541  xix  19:^^=1242  tons. 


1698-0 
808-8    2506 


Moment  about  L.w.li.  = 
„  ^  4  ^  19-25x4 


2506-8 


35 


=  2452  tons. 


Up  to  L.W.Ii. 
layer  l.wl.  to  3w.L. 


Tons. 

3311 
1242 


Moment 
about  iv.w.ii 


18862 


Displacement  to  3w.L.  =  2069  tons. 
16410 
.      c.B.  below  li.w.ii.  =  ^..^Q  =7-93'. 

4     19-25 
Tons  per  in. = 404-4  X  -x^^^^  =24-7. 


Transverse  bm= 
13-52'. 


3      420 
114291       4     19-25_ 
2069x35     3         3 


Explanation  of  Displacement  Sheet  (see  pp.  94,  95). 

The  length  of  the  ship  between  perpendiculars  is  divided  into 
twenty  equal  intervals,  and  the  immersed  depth  by  seven  equally  spaced 
water-planes,  the  lowest  being  2  feet  above  the  keel  amidships.  Below 
7  w.L.  is  treated  as  an  appendage,  it  being  preferable  in  all  cases 
not  to  take  the  lowest  w.l.  down  to  or  very  near  to  the  keel.  The 
ordinates  or  half-breadths  at  the  intersections  of  the  vertical  cross 
sections  with  the  horizontal  sections  are  measured  off  in  feet,  and 
set  down  in  dark  figures  (usually  in  red)  in  rows  opposite  their 
ordinate  number  and  under  their  w.l.  number.  Water-lines  that  are 
snubbed  or  cut  away  at  the  ends  should  be  produced  to  the  perpen- 
diculars by  eye  for  the  purpose  of  these  measurements  ;  the  volumes 
thus  added  are  afterwards  deducted  as  negative  appendages. 

The  Simpson's  multipliers  (halved  in  order  to  reduce  the  labour  of 
multiplication)  are  placed  against  the  ordinate  and  water-line 
numbers  ;  each  'ordinate  is  multiplied  by  the  multiplier  appropriate  to 
its  ordinate  number,  the  result  being  placed  on  the  right  ;  it  is  also 
multiplied  by  the  multiplier  appropriate  to  its  water-line,  and  the 
result  is  placed  underneath. 

Adding  the  former  products  in  columns  gives  the  functions  of  the 
water-planes  ;  these  are  multiplied  by  the  appropriate  water-line 
multipliers,  and  the  products  then  added,  giving  a  number  (3231"0  in 
the  text)  which  is  a  function  of  the  displacement.     The  displacement 


DISPLACEMENT  SHEET.  99 

ot  the  main  solil  is  obtained  by  multiplying  this  function  f  x  3^3  x 
spacing:  of  w.l.*s  x  spacing  of  ordinates  ;  the  factor  J  is  derived  from 
Simpson's  first  rule  applied  twice  in  succession,  tha  8  Is  2  for  both 
Bides  of  ship  x  4  for  the  half  multipliers  used  twice  instead  of  the 
whole  ones  ;   the  3^5  converts  cubic  feet  into  tons  (for  sea-water). 

The  functions  of  the  water-planes  are  again  multiplied  by  the 
number  of  intervals  from  the  l.av.l.  ;  the  sum  of  the  products 
(8706'78)  being  a  function  of  the  moment  about  the  l.w.l.  The 
multiplier  —  |  x  5^5  x  (spacing:  of  w.Ti.'s)  2  x  spacing  of  ordinates  — 
gives  this  moment  in  foot-tons. 

Again,  the  products  of  the  ordinates  with  the  water-line  multipliers 
are  abided  in  rows,  the  sums  being  functions  of  the  transverse  areas  ; 
these  are  multiplied  by  the  appropriate  ordinate  multipliers,  and  the 
products  added,  giving  the  same  function  of  the  displacement 
(3231*0)  as  before.  These  products  (headed  'multiples  of  areas ') 
are  further  multiplied  by  the  number  of  ordinate  spacings  from  amid- 
ships (station  11)  ;  the  products  are  added  for  each  end  of  the  ship, 
and  the  difference  between  the  suras  gives  a  function  (962'76)  of  the 
moment  of  the  main  solid  about  amidships.  On  using  the  multiplier 
g  X  g^5  x  spacing  of  w.l.'s  x  (spacing  of  ordinates)  2,  the  actual 
moment  is  obtained  in  foot-tons. 

The  lower  appendage  is  dealt  with  by  calculating  the  half-area  of 
each  transverse  section  below  the  lowest  w.l.,  ani  the  vertical 
position  of  its  e.g.  They  are  tabulated  on  the  left,  as  shown.  Each 
semi-area  is  multiplied  by  its  Simpson's  multiplier,  and  the  result  by 
the  number  of  intervals  from  amidships  ;  the  functions  of  areas  are 
also  multiplied  by  the  distances  of  their  e.g.  below  7  vi^.L.  The  three 
results  are  added  in  columns,  allowance  being  made  for  the  opposite 
signs  of  the  two  ends  of  the  longitudinal  moments,  the  sums  are 
converted  by  the  correct  multipliers  as  shown.  The  remaining 
appendages  are  calculated  by  the  ordinary  rules  for  volumes  and 
moments  of  solids,  rough  approximations  being  alone  required.  The 
'recess  '  is  that  due  to  the  emergo(nce  of  the  shafts.  A  table  is  set 
forth  containing  the  displacements  and  moments  of  each  item  ;  the 
total  displacement  and  the  position  of  the  centre  of  buoyancy  are  then 
found  by  simple  summation  and  division. 

To  obtain  the  position  of  the  longitudinal  metacentre  (see  p.  133), 
each  ordinate  of  the  load  water-plane  is  twice  multiplied  by  the 
number  of  intervals  from  amidships.  The  difference  between  the  sums 
of  the  first  products  for  each  ends  (326"53)  is  a  function  of  the 
moment;  the  multiplier — tx  (longitudinal  interval)2 — gives  the 
moment  of  the  main  portion  about  amidships.  The  sum  of  the  second 
products  multiplied  by  |  x  (longitudinal  interval)  3  gives  the  moment 
of  inertia  of  the  main  portion  about  amidships.  Area  of  the  main 
portion  is  obtained  from  the  function  of  area  (438"7)  multiplied  by  the 
multiplier  |  x  longitudinal  interval.  In  this  case  the  appendage  aft, 
being  fairly  large,  has  an  appreciable  effect  ;  its  area,  moment,  and 
moment  of  inertia  are  calculated  (the  last  being  equal  to  area  x 
(distance  of  e.g.  from  amidship3)2)  and  inserted  in  a  small  table  as 
shown.  Thus  the  total  area,  the  position  of  the  centre  of  flotation 
(or  e.g.  of  water-plane),  and  the  moment  of  inertia  about  11  are 
obtained.  The  correction  necessary  to  find  the  inertia  about  the  c.f. 
(see  p.  70)  is  then  introduced,  and  the  longitudinal  BM  =  moment  of 
inertia  about  c.f.  —■  volume  of  displacement.  By  assuming  an  approxi- 
mate vertical  position  for  the  e.g.  of  the  ship,  the  moment  to  change 

trim  I   inch   or   "^  ^  gm  (long.)    .^  obtained.     This,   together  with   the 

1-2  !> 
position  of  the  c.f.,  does  not  vary  greatly  with  moderate  changes  of 
draft  ;  and  they  are  generally  assumed  constant. 

The  remaining  particulars  evaluated  are  the  tons  per  inch  (equal 
to  area  of  water-plane  -r-  420),  the  area  of  midship  section  (equal  to 
function  of  area  for  11  x  g  x  water-line  interval),  and  the  transverse 
BM.  To  obtain  the  last,  the  cubes  of  the  ordinates  of  the  water-plane 
are  multiplied  by  Simpson's  multipliers,  and  the  products  added.     The 


DISPLACEMENT  SHEET.  101 

•am  multiplied  by  ^  x  lontjitudinal  irterval  '.s  equal  tj  thj  n^omtrb  of 
inertia  of  the  water-plane  about  amidships  ;  on  dividing  this  by  the 
volume  of  displacement,  the  transverse  bu  is  obtained  (see  p.   111). 

Frequently  the  displacement,  tons  per  inch,  transverse  bm,  and 
vertical  position  of  the  c.b.  are  required  for  other  water-lines.  Here 
they  are  worked  out  for  'a'  w.l.  <2  feet  above  l.w.l.),  2  w.l.,  and 
Sw.L.  The  process  consists  of  finding  the  volumes  and  moments  of 
the  layers  between  the  respective  w.l.  and  the  l.w.l.;  and  adding  to, 
or  subtracting  from,  the  displacement  and  moment  for  the  l.w.l. 
The  multipliers  used  at  a. w.l.  are  6,  8,  —1  for  volumes,  and  7,  6,  — i  for 
momenta  about  middle  ordinate  (see  pp.  46  and  68).  For  2w.L>.  the 
multipliers  are  5,  8,-1  for  volumes,  and  3,  10,-1  for  moments  about 
end  ordinate.  For  3  w.li.  the  ordinary  Simpson's  multipliers  are  employed. 
When  the  after  appendage  is  large,  it  is  desirable  to  use  the  tons  p«r  inch 
and  C.F.,  both  corrected  for  after  appendage,  instead  of  the  "  functions  of 
area"  taken  on  p.  97.  The  transverse  BM's  are  obtained  by  cubing  the 
ordinates  as  for  the  li.w.L. 

Explanation    o?    Displacement    Sheet    using    Tciiebycheff's    Eule. 

The  horizontal  water-lines  are  spaced  equidistantly  as  with  the 
preceding  displacement  sheet  ;  an  appendage  is  left  below  7  w.l.  The 
vertical  transverse  sections  are  spaced  so  as  to  meet  the  requirements 
of  Tchebycheff's  rule  (see  p.  46),  using  five  ordinates  for  each  half 
of  the  length,  i.e.  ten  ordinates  in  all ;  the  positions  of  the  sections 
are  indicated  at  the  top  of  the  sheet.*  The  ordinates  are  measured 
from  the  half-breadth  plan  ;  they  are  numbered  I,  II,  III,  IV,  V  for 
the  fore  end,  commencing  from  amidships,  and  I  a,  II  a,  III  a,  IV  a, 
Va  for  the  after  end,  commencing  from  amidships.  The  half-breadths 
are  measured  off  in  feet  and  inserted  in  the  table  against  the  number 
of  the  corresponding  ordinate,  and  under  the  corresponding  water- 
line,  in  dark  figures  (usually  in  red).  Under  each  water-line  is  set 
the  correct  Simpson's  multiplier,  halved  for  convenience  ;  no  multi- 
plier is  required  opposite  the  ordinates. 

The  ordinates  are  first  added  in  columns,  the  sums  being  functions 
of  areas  of  the  water-planes.  These  are  mult4plied  by  the  corre- 
sponding Simpson's  multipliers  and  their  sum  (1076'82)  is  a  function 
of  the  displacement  ;  the  multiplier  required  to  obtain  the  displace- 
ment   in    tons    is    2    (for    both    sides)  x  §  (Simpson's    rule    for    half 

multipliers)  x  ^  (salt  water)  x  water-line  spacing —^ — .  The  pro- 
ducts of  the  functions  of  the  water-line  areas  are  also  multiplied  by 
the  number  of  intervals  from  the  l.w.l.;  the  results  are  functions  of 
vertical  moments.  The  sum  of  these  multiplied  by  |  x  g^j  x  (water- 
line  spacing)  2  x    ^°^ — is   the   moment  of  the  main  solid  about   the 

L.W.L. 

Each  ordinate  is  afterwards  multiplied  by  its  appropriate  water- 
line  multiplier,  and  the  products  added  in  rows  ;  the  sums  are 
functions  of  the  areas  of  the  transverse  sections.  The  sum  of  these 
gives  a  function  of  the  displacement,  which  should  be  the  same  as 
that,  previously  obtained  (1076*82).  The  differences  between  the 
functions  of  the  areas  for  the  fore  and  after  ends,  taken  in  pairs,  are 
written  down  and  multiplied  by  the  levers  equivalent  to  the  Tcheby- 
cheff  spacings  expressed  in  terms  of  half  the  length  ;  in  the  example 
the  functions  for  the  after  body  are  greater  in  each  case  than  the 
corresponding  ones  for  the  fore  body,  but,  if  this  is  not  the  case, 
allowance  should  be  made  for  sign.  The  products  are  functions  of 
moments  about  amidships  ;  their  sum  multiplied  by  $  x  ^  x  water- 
line  spacing  x ~ is  the  moment  of  the  main  solid  about  amid- 
ships expressed  in  foot-tons. 

Unless  a  special  body  has  been  constructed  with  Tchebycheft 
sections,  the  calculation  of  the  lower  appendage  is  the  same  as  that 
in  the  ordinary  displacement  sheet,  equidistant  sections   being  used. 

*  Alternatively  the  ordinary  *  Simpson  '  sections  numbered  2,  5,  7, 10, 
12. 15, 17  and  20,  may  be  taken  instead  of  the  exact  sections  required  for 
Tchebycheff's  rule  with  four  ordinates  repeated. 


102 


WEIGHT  AND  CENTRE  OF  GRAVITY. 


The  lertiainiag  appendages   are   oalqulated  as   before;    the  final  table 
has  not  been  inserted  in  t&is  case.    ' 

The    calculation    for    the    transverse    metacentre    is   the    same    as 

before  except  that  the  cubes  of  the  ordinatea  are  added  direct,  and 

their  sum  multiplied  by  —  x  ■  ^"^*      to  obtain  the  transverse  moment 

o  ID 

of   inertia. 

For  the  cenire  of  flotation,  the  differences  of  the  ordinates  in  pairs 
are  written  down,  allowing  for  sign  if  necGssary,  and  multiplied  by 
the  Tchebycheff  levers.  The  sum  of  the  products  is  a  function  of 
moments  ;  and  the  distance  of  the  c.f.  abaft  amidships  is  equal  to 
function  of  moments  ^  length.      „,  ,.  ,^1  .        ,         . 

— z r. 1 X  ■ — ,r—        The  area  of  the  water-plane  is  equal 

function  of  areas  2  f  ^ 

to  function  of  area    x  2  x  ^"^    '     To  obtain  the  longitudinal  moment 

of  inertia,  the  ordinates  are  added  in  pairs,  and  multiplied  by  the 
squares  of  the  Tchebycheff  levers  ('007,  '038,  '25,  -472,  -840).  The  sum 
of  the  products  multiplied  by  a  twentieth  of  the  cube  of  the  length 
is  the  moment  of  inertia  about  amidships.  The  corrections  for  the 
after  appendage,  and  for  the  position  of  the  c.f.,  and  the  calculation 
of  BM  are  similar  to  those  on  the  ordinary  sheet. 
The  remaining  calculations  are  made  as  before. 


WEIGHT  AND  CENTRE  OF  GRAVITY  OF  SHIPS. 

In  the  early  stages  of  design,  an  approximation  must  be 
made  to  the  weight  of  a  ship  in  order  to  determine  whether 
it  is  equal  to  the  displacement  assumed  ;  the  position  of  the 
centre  of  gravity  is  also  required  in  order  to  determine  the 
stability   and    trim. 

The  weight  of  a  ship  is  conveniently  divided  into  six 
items  :  hull,  equipment,  machinery,  fuel,  armour,  and  arma- 
inent.  In  a  merchant  vessel  the  two  last  named  are  replaced 
by  the  load  to  be  carried.  The  proportions  vary  greatly  in 
different  ships  ;  those  in  the  table  are  illustrative  of  certain 
types  : — 


Type  of  Ship. 

Percentage  Weight. 

i 

>> 

g 

c 

^d 

u 

Armament  or 

"5 

'3 

-§ 
^ 

0 

Load. 

W 

H 

^ 

^ 

-  < 

Battleship       .     . 

34 

3 

10 

3^ 

31J 

18 

First-class  Cruiser 

38 

4 

17 

7 

20 

14 

Light  Cruiser.     . 

43J 

6 

20 

12 

12^ 

6 

T.   B.    Destroyer 

(deep  condition) 

34 

4 

34 

25 

— 

3 

Steam  Yacht  .     . 

61 

10 

19 

10 

— 

— 

Atlantic  Liner     . 

55 

included 

27 

16 

— 

2  (Passengers 

in  hull. 

and  Stores) 

WEIGHT    OF    HULL. 


103 


Hlll. 
First  Estimate. 
The  weight  of  hull  is  determined  to  a  first  approximation  * 
in  a  variety  of  ways.     In  ships  of  very  similar  typo  it  may 
bo  assumed  to  be  the  same  percentage  of  the  displacement, 
e.g.  340/0   in  battleships,  etc.     Or  it  may  be  compared  with 
the   product   length  X    (breadth  +  depth)  amidships,   the  co- 
eflBcient  being  determined  from  a  similar  ship,  making  allow- 
ance for  any  great  alteration  of  scantlings.     Mr.  J.  Johnson 
(in  Trans.  Inst.  Nav.  Archs.,  1897)  published  a  useful  method 
for  approximating  to  the  hull  weight  of  a  vessel   built  to 
the  highest  class  at  Lloyd's  or  Veritas. 
If  N  =  a  modification  of  Lloyd's  old  longitudinal  number 

=  Length  from  after  part  of  stem  to  fore  part  of  stern  post 
on  upper  deck  beams   x  {i  greatest  moulded  breadth 
+  depth  from  top  keel  to  top  upper  deck  beams  + 
^  midships  girth  to  upper  deck  stringer}. 
In  spar-  and  awning-deck  vessels  the  girths  and  depths  are 
measured  to  the  spar  or  awning  decks  ;    they  are  taken  to  the 
main  deck  in  one-,  two-,  and  well-decked  vessels, 
w  =  finished  weight  in  tons  of  the  steel  hull. 

=  K  (-^)  ^ 

\ioo  / 


Then  w 


or  log  10  W  =  X  log  10  (-3^)  -  ^• 
Where  x  and  K  or  A  are  determined  from  the  table  below : — 


Type  of  Vessel, 


Three  deck       .... 

Spar  deck 

Awning  deck    .... 
One-,  two-,  and  well-deck 
Sailing 


1-40 
1-35 
1-30 
1-30 
1-40 


0-492 
0-576 
0-665 
0-856 
0-410 


0-308 
0-240 
0-177 
0-068 
0-387 


The  distance  of  G  abaft  the  middle  of  length  and  above 
the  keel  can  be  estimated  from  information  available  for 
other  ships,  taking  these  distances  proportional  respectively 
to  the  length  and  totsd  depth. 

Detailed  Estimate. 

In  the  later  stages,  when  scantlings  are  fixed,  the  weight 
and  centre  of  gravity  of  hull  are  found  as  follows: — 

The  hull  may  be  divided  into  two  groups  :  (1)  calculable 
items  forming  about  60  0/0  (in  a  warshij))  of  the  whole,  which 
include  the  greater  part  of  the  structure,  and  (2)  *  judgment ' 
items,  including  portions  of  complicated  structure  and  fittings. 

The  latter  can  only  be  assessed  by  comparison  with  known 
weights   in   a   similar    ship    (recorded   weights   if   possible)  ; 
the  centre  of  gravity  of  each  item  can  usually  be  determined 
♦  See  also  under  "  Design  ". 


104  WEIGHT    OF    HULL. 

with  fair  accuracy  from  its  position.  The  former  are  directly 
calculated  from  the  scantlings  ;  the  manner  of  so  doing  is 
indicated  in  a  few  instances  below.  To  each  item  3o/o  should 
be  added  for  fastenings  (or  such  an  addition  should  be  made 
at  the  end  of  the  calculation). 

If  the  stresses  on  the  ship  are  also  required,  ifc  is  con- 
venient  to  divide  every  item  into  portions  wholly  before  and 
wholly  abaft  the  midship  section.  Moments  are  taken  about 
two  fixed  planes,  one  being  generally  the  midship  section 
and  the  other  the  l.w.l.  or  the  keel. 

Outer  Bottom  Plating. — Assume  all  of  uniform  thickness, 
repeating  as  necessary  for  the  portions  whore  the  thicknesg 
is  in  excess  or  in  defect  of  that  assumed.  Divide  the  length 
into  sections  spaced  equidistantly,  and  measure  the  half-girthp 
at  each  section.  Apply  the  method  of  Rule  II,  par.  24,  p.  58, 
obtaining  the  '  modifying  factor  '  at  each  section.  (This  is 
the  ratio  of  the  slant  length  of  the  mean  water-line  intercepted 
between  the  sections  to  their  perpendicular  spacing.)  Find 
the  height  of  the  centre  of  gravity  of  each  section  by  dividing 
it  into  four  equal  parts,  and  proceeding  as  in  par.  16,  p.  62. 
Then  arrange  the  calculations  as  in  the  following  table  for 
the  forward  portion  of  a  warship  below  the  armour  deck,  the 
percentage  for  laps,  butts,  and  liners  being  taken  as  calculated 
for  an  average-sized  plate  (say  20'  x  4'). 

Transverse  Framing. — There  are  usually  several  varieties, 
such  as  web  and  ordinary  frames,  or  bracket,  lightened  plate, 
and  watertight  frames.  To  avoid  calculating  each  one 
separately,  calculate  the  weight  and  height  of  e.g.  for  a 
specimen  frame  at  intervals  of  about  ^^  length.  Plot  these  as 
curves  on  a  base  of  length  of  ship,  drawing  separate  curves 
for  each  type  of  frame.  The  weight  and  e.g.  position  can 
then  he  read  off  for  each  frame,  or  a  mean  can  be  taken  for 
a  group  of  similar  frames  coming  together. 

Longitudinal  Framing. — Usually  uniform  in  section,  but 
the  height  of  e.g.  must  be  taken  at  equidistant  intervals, 
and  the  mean  taken. 

Bulkheads. — Main  bulkheads  are  usually  thicker  towards 
the  bottom,  and  the  e.g.  is  below  the  centre  of  area.  Take 
the  minimum  thickness  of  plating  and  calculate  its  weight 
and  e.g.  without  any  allowance.  Then  add  the  additional 
thickness,  the  stiffeners,  and  allowances  for  laps,  butts,  and 
fastenings  ;  all  but  the  first  item  have  the  e.g.  at  centre  of 
area.  Then  find  the  ratio  of  the  initial  to  final  weights,  and 
of  the  e.g.  below  centre  of  area  to  the  height  of  bulkhead. 
If  this  be  done  for  one  or  two  typical  bulkheads,  the  rest  may 
be  determined  from  the  simpler  first  calculation  by  allowing 
the  same  ratios.  The  ratio  of  weight  for  the  main  bulkheads 
of  a  warship  is  1  :  1'9  ;  for  ordinary  below-water  bulkhead/* 
it  is   1  :  1  66. 


WEIGHT    OF    HULL. 

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106  WEIGHT    OF    HULL. 

For  smaller  bulkheads  between  decks,  measure  total  length 
and  multiply  by  mean  distance  between  decks.  From  this 
area  the  weight  of  plating  is  at  once  found  ;  12|ob  is  a  usual 
percentage  to  add  for  laps.  The  length  of  the  boundary 
bars  is  readily  determined  ;  that  of  the  stiffeners  is  found 
by  dividing  the  area  by  their  spacing,  adding  a  percentage 
as  necessary  for  brackets  at  heads  and  heels. 

Decks. — Take  in  sections,  each  section  having  uniform 
thickness.  Find  area  and  longitudinal  e.g.  of  each  portion. 
Weight  of  plating  is  found  as  with  bulkheads  ;  that  of  beams 
is  equal  to  area  X  weight  of  beam  per  foot  run  -f  beam 
spacing,  adding  a  small  allowance  for  beam  knees  or  brackets. 
For  planking,  find  volume  of  wood  and  multiply  by  its 
density,  allowing  about  60/0  for  fastenings,  e.g.  a  3  in.  deck 
weighs  14  lb.  per  square  foot  if  of  teak,  12  lb.  if  of  fir, 
including  fastenings. 

When  all  the  items  for  the  hull  have  been  evaluated,  they 
are  tabulated  in  the  manner  shown  in  the  succeeding  pages. 
The  calculations  therein  given  are  those  for  a  paddle  tug 
145'  X  28'  X  11'  4"  (4    feet  freeboard)  x  750  tons  displacement. 


SUMMAKY   OF  WEIGHT   OF  HULL. 
Note. — Base  for  vertical  C.G.s  top  of  bar  keel. 


Item; 


Tons. 


Outer  bottom  plating  .     . 

Bar    keel 

Stem 

Sternpost 

Rudder,  etc 

Intercostal  M.L.  keelson 
Transverse  bulkheads .  . 
Transverse    framing     (ex 

webs) 

Cant  frames 

Peep     beams     and     web 

frames 

Side    keelsons    and    bilge 

stringers      ..... 
Top  of  reserve  feed  tank 

Upper   deck 

Lower  deck  forward .  . 
Lower  deck  aft  .  .  . 
Bridge  deck  .... 
Sponsons,  etc.  .... 
Paddle-boxes  .... 
Coal-bunker  bulkheads  . 
Division  ia  chain  locker 
Divisions    in    fresh-water 

and   ballast    tanks    .     . 

Tops ides  

Main  engine  bearers  .  . 
Auxiliary  engine  bearers 
Boiler    bearers    .... 

Carried  forward  .    . 


64-5 

3-5 

•4 

•4 

1-7 

2-5 

15-0 

27-0 
•6 

10-0 

8-5 
2-3 

as-0 

5-0 

4-5 

10-0 

12-5 

12-0 

7-3 

•3 

•6 

4-5 

14-5 

3-0 

1-5 


250- 1 


Before  6. 


C.G    Mmt. 


72 


2-0 

55 


Abaft  6. 


C.G    Mmt. 


28-8 


7-5 


41-4 

220  0 

(56  0 
31-2 
30-0 

16-5 


2030 
9-0 


20 


72-6 
740 


38 

75 

1-1 
30 
3^5 

52 

19 


191 


12i)0 


29-0 
125-8 


102-6 
450 

110 

25-5 

1330 

2310 

138-7 


21-2 
27-0 


28- 


1050-4 


Above  base. 


O.G.     Mmt 


6-7 

•33 

10-6 

8-& 

ll-O 

■9 

90 

50 
16-0 

9-5 

34 

2-0 

160 

8-7 

8-5 

22-6 

135 

190 

7-9 

4-0 


18-1 
4-5 

1-8 
2-1 


432-0 
1-2 
4-2 
34 
18-7 
2-3 

1350 

135-0 
9-6 

950 


4-6 

6080 

43-5 

38-2 

2260 

168-8 

2280 

67-7 

1-2 

32 

81-5 

652 

64 

3-1 


2397-3 


WEIGHT    OF    HULL. 


107 


Item. 


Brought  forward. 
Houses    and    fittings    on 

and   above    bridge 
Sponson  houses 

Rubber  

Companions    .     .     . 
Skylights    .... 
Engine  and  boiler  casings 
Steering    gear      . 
Ventilation     .     . 
Pumping     .     .     . 
Pillars         .     .     . 
Cathead       .     .     . 
Mooring  pipes  and  chock 

Horn 

Towing  hooks  andstilTen 

ing 

Towing   beams    and   sup 

ports 

Bollards  and  fairleads 
Samson  post  .  .  . 
Boats  davits  .  .  . 
Awning   stanchions,    etc 

Ladders 

Fittings    in    galley 
W.O.    fittings      .     .     . 
Fittings  in  other  sponson 

houses  .  .  . 
Miscellaneous  upper  deck 

fittings  .  .  . 
Shovelling  flat  . 
Side  scuttles  forward  and 

aft  .... 
Side  scuttles   to  sponson 

houses     

Coaling  scuttles  .  . 
Engineers      store  -  room 

bulkhead      .... 
Cabin  bulkheads  forward 
Cabin  bulkheads  aft 
Corticene  on  lower  deck 

aft 

Silicate    lagging 
Chain  locker  fittings 
Fittings  in — 

Fore  crew  space 

After  crow  space 

Cabins    forward    . 

Cabins  and  mess  aft 

Cabin  and  saloon 

Fore  holds        .     . 

After  holds    .    .     . 

Engineers'  Stores 
Topside    fittings 
Cement        .... 
Paint      


Tons, 


250- 1 

3-5 

4-5 

10-0 

2'j> 

2-0 

9-0 

•1-5 

1-2 

1-5 

•5 

•4 

3-2 

•4 

11 

•9 
12-0 

•5 
2-2 

•8 
1-0 
1-2 

•6 


60 
1-2 


1-0 


Before  6. 


C.G.    Mmt. 


15-6 
2-5 


64 


1-0  55 
20 


2-5 
16 


20 


1-6 
33 


55 


347-4 


45 


653-4 


54-2 
112 


120 
24-4 

29-2 


320 


Abaft  6.     I  Above  base. 
C.G.   Mmt.    C.G.  I  Mmt, 


2 
2 
•5 

11-8 

22  0 

50 

5-0 


250 


22 


20 


1-9 
16-5 


27-6 

650 
760 
19-2 


67-5 


IO08O 


130 


35 
18-3 


18 


48 


50 
367 


48 


214 


1050-4 


200 

5-0 

1-0 

106-2 

330 

60 


2-5 


27-5 


51-3 
900 


48-4 
15-6 


21-0 
21-9 


10-8 


43-2 


200 
110 


f6  0 

870 


720 
240 


1831-3 
10  8  0 

746-3 


270 
18-5 
150 
18-7 
21-5 
190 
16  5 
18  0 
10-0 
12  0 
21-0 
190 
220 

190 

20-5 
165 
180 
22  0 
22-5 
150 
17-0 
170 


17-0 

165 
1-5 

140 

20-0 
15-3 

65 
12 

12 

8-5 

9 

3 

12 

12 

12 

12 

12 
56 
55 
7-0 

180 
1-1 

10 


110 


C.G.  abaft  6,  2  14'.    Above  base,  11'. 

Take  350  tons.    2  1'  abaft  6.    11'  above  base. 


108  WEIGHT  OF  EQUIPMENT,  ETC. 

Equipment. 

This  is  conveniently  divided  (in  warships)  as  follows  : 
Fresh  water  (for  10  days  allow  about  8*7  tons  for  100  men) ; 
provisions  and  spirits,  including  tare  (for  4  weeks  allow 
about  5*7  tons  per  100  men) ;  officers'  stores  and  slops  ; 
officers,  men,  and  effects  (allow  8  to  the  ton) ;  masts,  rigging, 
sails,  etc.  ;  cables  (500  fathoms)  ;  anchors  ;  boats  ;  warrant 
officers'  stores    (4  months) ;    torpedo   net  defence. 

In  passenger  ships  undergoing  long  voyages  allow  1  tonj 
per    5     persons    for    passengers'    gear,    including    baggage,- 
bedding,   etc.,   also   '025   ton   por  day   per  person   (average) 
for  water  and  provisions. 

Machinery  and  Fuel. 

In  the  preliminary  estimate  the  weight  of  machinery  is 
based  on  the  total  power.  Coefficients  for  various  types  of 
machinery  are  given  on  pp.  389,  390.  Information  obtained 
from  actual  ships  should  also  be  utilized  where  possible. 

The  weight  of  coal  assumed  is  frequently  an  arbitrary 
amount  less  than  the  full  bunker  capacity.  The  full  coal 
storage  can  be,  however,  determined  from  the  volume  of 
the  bunkers  calculated  by  the  rules  on  p.  54,  the  areas  of 
the  sections  being  taken  to  underside  of  beams  only.  About 
10  or  15  per  cent  (varying  with  type  and  shape  of  bunker) 
is  then  deducted  for  broken  stowage  ;  the  net  volume  in  cubic 
feet,  on  being  divided  by  43  (North  Country  coal),  40  (Welsh 
coal),  36  (patent  fuel  symmetrically  stowed)  or  45  (patent 
fuel  shot  into  bunkers),  gives  the  stowage  in  tons. 

The  weight  of  liquid  fuel  is  equal  to  the  whole  volume 
in  cubic  feet  divided  by  38-5. 

In  all  cases  the  centre  of  gravity  of  th3  fuel  is  the  e.g. 
of  the  volume  (see  p.  66). 

Armour. 

The  weight  and  position  of  e.g.  of  armour  in  warships  are 
determined  by  a  process  similar  to  that  adopted  for  the  hull. 
If  the  armour  is  not  specified  by  its  weight  par  square  foot 
of  plate,  this  can  be  determined  from  its  thickneS(3>  sim^a  it 
weighs  4951b.  per  cubic  foot.  Add  l^o/o  for  bolts.  Backing 
is  dealt  with  similarly  to  the  planking  of  a  teak  deck. 

Armament  or  Load. 

The  weight  of  guns,  mountings,  charges,  and  projectiles 
are  known  (see  pp.  380-385),  and  the  position  of  the  e.g. 
of  each  item  can  generally  be  spotted  without  difficulty. 
Allow  30  to  40%  tare  for  cartridge  cases. 


SUMMARY    OF    WEIGHTS 


109 


The  load  in  a  cargo  ship  is  generally  determined  before- 
hand. Its  e.g.  is  usually  found  by  assuming  the  whole  space 
available  to  be  filled  with  a  homogeneous  cargo — the  assump- 
tion the  most  unfavourable  to  the  stability  ;  the  e.g.  is 
then  that  of  the  volume  of  the  hold. 

For  passengers  without  baggage  allow  16  to  the  ton  with 
men,  women,  and  children  ;  14  to  the  ton  with  men  only.  In 
pleasure  steamers  where  the  stability  can  be  affected,  assume 
the  e.g.  of  passengers  seated  to  be  6  inches  above  the  seat  ; 
for  those  standing?,  2  feet  above  the  deck  is  generally  a  safe) 
assumption. 

The  final  weight  and  position  of  centre  of  gravity  are 
found  by  adding  together  the  weights  and  moments  of  the 
several  portions  as  shown  in  the  table  below  :  — 


Summary  OF  Battleship  Weights  (580'  x  9C 

)'x2r6"- 

-44'deep). 

Moment 

Moment 

Moment 

Item. 

Weight. 

from 

above 

below 

amidships. 

li.W.Ii. 

li.W.L. 

Weight    before   amid- 

Tons. 

Ft.-tons. 

Ft.-tons. 

Ft.-tons. 

ships— 

Armament  .... 

2076 

236110 

43020 

1870 

Armour 

3270 

353300 

42077 

1901 

Hull 

3770 

355340 

25473 

28796- 

Machinery  .... 

820 

29500 

— 

8200 

Coal 

670 

23000 

1360 

5840 

Equipment  .... 
Weight    abaft    amid- 

400 

55000 

7420 

766 

1052250 

ships — 

Armament  .... 

2763 

348020 

46280 

2190 

Armour 

3585 

481250 

28848 

3563 

Hull 

4933 

697170 

24849 

38897 

Machinery  .... 

1930 

198800 

— 

19300 

Coal 

330 

21400 

640 

3060 

Equipment.     .     .     . 

335 

48200 

3564 

52 

1 

24882 

1694840 
1052250 

223531 
114435 

114435 

1 

642590 

109096 

Total  weight  of  ship- 

=  24882  ton 

s. 

C.G.  abaft  y^  -  642590  _ 
^         24882 

7  feet. 

C.G.  above  L.W.L.  -  ^^f ^^6  - 

4-38  feet 

24882 

110 


STABILITY. 


Fig.  120 


STABILITY. 

If  a  ship  be  slightly  disturbed  from  a  position  of  equi- 
librium, and  if  the  forces  then  in  operation  tend  to  restore  the 
original  position,  the  equilibrium  is  termed  stable  ;  if  the 
forces  tend  to  move  it  further  from  the  original  position, 
the  equilibrium  is  termed  unstable  ;  if  it  shows  no  tendency 
to  move  away  from  or  return  to  the  original  position,  the 
equilibrium  is  termed  neutral. 

The  equilibrium  of  a  ship  is  always  stable  as  regards 
vertical  deflections  causing  an  alteration  of  displacement  ; 
the  only  disturbances  that  need  examination  consist  of  inclina- 
tions about  horizontal  axes  with  the  displacement  unaltered. 
Of  these  the  principal  are  :  (a)  in'^lination  in  a  transverse 
plane  about  a  longitudinal  axis,  and  (b)  inclination  in  a 
longitudinal  plane  about  a  transverse  axis.  The  stability  in  these 
directions  is  termed  transverse  and  longitudinal  respectively. 
Transverse  Stability. 
T^ig.  119  is  a  transverse  section  of  a  ship  heeled  over 
through  a  certain  angle  6.  w'l'  is  the  water-line  for  the 
inclined,  position,  and  wl  is  the  water-line  for  the  upright 

position.  These  two 
planes  intersect  each 
other  in  a  longitudinal 
direction,  and  bound  two 
2  wedges  l'sl  and  wsw 
equal  in  volume  to  each 
other,  provided  the  dis- 
placement remains  the 
same.  The  wedges  are 
.  ,  called  respectively  the 
wedges  of  immersion 
and  emersion,  or  the  in 
and  out  wedges.  G  is 
the  centre  of  gravity  of 
the  ship  and  b'  her  centre 
of  gravity  of  displace- 
ment, or  centre  of  buoyancy.  The  weight  of  the  ship  then 
acts  vertically  downwards  through  G,  and  the  resultant 
pressure  of  the  water  acts  vertically  upwjards  through  b', 
these  two  forces  forming  a  righting  couple,  the  arm  of  which 
is  GZ — that  is,  the  perpendicular  distance  between  the  lines 
of  action  of  the  two  forces.  The  moment  of  this  couple— that 
is,  the  weight  of  the  ship,  or  its  displacement,  multiplied  by 
the  length  of  the  arm  GZ— is  the  moment  of  statical  stability 
of  the  ship  at  the  given  angle  of  inclination  e.  This  moment 
is  generally  expressed  in  foot-tons — that  is,  the  weight  of  the 
ship  in  tons  multiplied'  by  the  length  of  the  arm  Gz  in  feet. 
B  is  tho  centre  of  buoyancy  of  the  ship  when  upright  ;  S^  ig 
the  point  of  intersection  of  the  two  water-lines,  I  the  point 
where  the  vertical  b'm  cuts  the  plane  of  flotation  ;  g  and  g' 


^IG.  119 


STABILITY.  Ill 

are  the  centres  of  gravity  of  the  emerged  and  immersed  wedges 
respectively,  gh  and  g'h'  being  perpendiculars  dropped  to 
g  and  g'  from  the  plane  of  flotation  w'l'.  The  point  M.  where 
the  vertical  line  BM,  drawn  through  the  centre  of  buoyancy 
B  when  the  ship|  is  in  (an  upright  poaition,  cubs  the  vertical" 
line  B'M,  drawn  through  the  centre  of  buoyancy  b'  for  the 
inclined  position,  is  termed  the  tranwerse  metacentre  when  the 
ship  is  inclined  through  an  indefinitely  small  angle,  and  also 
when  the  point  of  intersection  is  the  same  for  all  angles  of 
heel. 

If  the  centre  of  gravity  g  is  below  the  metacentre  m,  the 
equilibrium  is  stable  ;  if  a  is  above  M,  the  vessel  is  unstable, 
and  will  capsize  or  at  least  h^ol  to  a  larga  angle  ;  if  o 
coincides  with  M,  the  equilibrium  is  neutral. 

The  intersection  of  the  new  vertical  through  b'  is  found 
usually  to  pass  very  near  the  metacentre  m  for  all  angles  of 
heel  Tip  to  10°  or  15°.  Within  these  limits  the  stability  lever 
GZ  is  equal  to  GM  .  sin  Q  \  or  the  moment  of  statical  stability 
ia  w  .  GM  sin  Q. 

For  moderate  angles  the  stability  depends  wholly  on  the 
value  of  GM,  which  is  termed  the  metacentric  height.  The 
position  of  G  is  calculated  by  the  rules  given  on  pp.  102-9  ; 
that  of  M  is  obtained  by  the  process  indicated  below. 

To  obtain  the  height  of  the  transverse  metacentre. 

Assume  the  angle  of  heel  B  to  ^ig.  121. 

be  small,  let 

y  =  half  breadth  WS  or  SL  at 
any  station. 

V  =  volume  of    either    wedge 

wsw'  or  lsl'. 
j7,  <7i  =  c.g.s.  of  the  wedges. 
h,  h\  =<eet  of  perpendiculars 

from  g,  g\  on  wsl'. 

V  =  volume  of  displacement. 

I  =  moment  of  inertia  of  water- 
plane  about  longitudinal 
axis  through  S. 

dx  =  an  element  of  length  of 
ship. 

Then  BR  =  ^  .  hhi. 

Also  V  .  hhi  =  moment  of  transference  of  wedges. 


'■P- 


2       3 

=  16  \  y^dx  =  0.1. 

.'.  BR  =  »  .  I/V. 
Also  BR  =  6  .  BM  approximately 

Whence  BM  =  ~ 


.  dx  approximately. 


112  METACENTRIC   DIAGRAM. 

The  height  of  the  transverse  metacentre  above  the  centre  oi 
buoyancy  is  equal  to  the  moment  of  inertia  of  the  water-plane 
area  about  the  axis  of  inclination  (in  this  case  the  centre  line) 
divided  by  the  volume  of  displacement. 

The  actual  calculations  for  the  transverse  bm  at  several 
draughts  of  a  ship  are  given  in  the  displacement  sheets  on 
pp.  94,  100.  The  moment  of  inertia  I  is  there  expressed  by  the 
integral  ^^y^  .dx;  the  cubes  of  the  ordinates  are  first  integrated, 
and  the  result  muliplied  by  the  factor  §. 

In  a  ship  whose  sections  are  circular  in  the  neighbourhood 
of  the  water-line,  such  as  a  submarine,  the  metacentre  is 
coiucident  with  the  centre  of  the  circular  arcs. 

.Definition. — The  surface  stability  of  a  ship  is  that  obtained 
when  the  centre  of  gravity  coincides  with  the  upright  centre 
of  buoyancy. 

Metacentric  DiAGRA]\r. 

The  stability  of  a  ship  in  various  conditions  is  conveniently 
exhibited  by  means  of  a  metacentric  diagram.  In  fig.  122, 
which  shows  the  diagram  for  the  ship  taken  in  the  displace- 
ment sheets  (pp.  95,  97),  and  inclining  experiment  (pp.  135, 
138),  the  vertical  scale  of  draught  is  intersected  by  a  straight 
line  drawn  at  an  angle  of  45°.  From  the  intersection  of 
'a.'w.l.,  L.W.L.,  2  W.L.,  and  3  w.l.,  with  this  line  are  set  up 
(or  down)  the  vertical  positions  of  the  centres  of  buoyancy, 
and  of  the  metacentres  ;  these  being  obtained  from  the  dis- 
placement sheets.  Through  the  spots  thus  obtained  are  drawn 
the  curve  of  metacentres  and  curve  of  buoyancy,  giving  the 
positions  of  m  and  B  at  intermediate  water-lines. 

The  heights  of  the  e.g.  are  calculated  for  a  number  of  condi- 
tions of  the  ship  ;  they  are  here  shown  for  the  inclining 
condition  (see  inclining  experiment,  p.  135),  the  legend  (or 
normal)  condition,  deep  load  condition,  and  light  condition. 
A  cargo  or  passenger  ship  is  frequently  worked  out  for  a 
large  number  of  conditions  as  regards  stowage  of  cargo,  coal, 
and  water  ballast.  From  the  curves  the  position  of  the 
metacentre  at  any  water-line  is  obtained,  and  the  vertical 
position  of  G  marked  on  ;  the  metacentric  height  is  thus 
determined  and  recorded. 

Value   of   gm   in  Typical  Ships. 

A  vessel  having  a  low  metacentric  height  is  termed  cranio  ; 
one  provided  with  a  large  gm  is  termed  stiff.  A  crank  vessel 
usually  rolls  less  and  moves  more  easily  among  waves  than 
a  stiff  vessel  ;  for  this  reason  the  value  of  the  gm  adopted  for 
a  ship  is  made  as  small  as  possible,  consistent  with  safety 
and  other  considerations.  Typical  values  are  given  in  the 
following  table  : — 


METACENTRIC   DIAGRAM. 


113 


Fjg.  102. 

Metacentric  I 

)iAGRAM  OP  Small  Cruiser. 

\v.                                 Curve  of 

MEAN     TONS 

TONS 

^->^                   ^k^ctRcenties. 

^«AU6HTp^,'^S-, 

PER 

■p- 

p-^-— 

MENT. 

~co 

, 

~00 

00 

oc 

l6-0>i 

4.140 

A 

CO 

CO 

CM 

CM 

/ 

15-6 

3.964 

27  65 

Z 

X 

Z 

^ 

14-- 6 

3.623 

2714 

B 

. 

13-6 

3.303 

26-65 

/ 

/ 

12-7 

3  031 

C 

1                                                        f                        'y 

^2-3 

2.947 

D 

/ 

11-6 

2.677 

2615 

/ 

9-6 

2,067 

24  81 

/ 

> 

c 

urvet 

Jf 

^ 

^ 

bu 

oyanc 

y. 

A.  Deep  condition.    Coal  725,  oil  142,  reserye  feed  92  tons. 

B.  Normal  condition.     Coal  450  tons. 

C.  Light  condition.    No  coal ;  no  consumable  stores. 

D.  Condition  as  inclined. 


Note. — The  curve  of  buoyancy  is  generally  nearly  straight  ; 
4he  tangent  of  its  inclination  to  the  horizontal  is  equal  to 

12  X  depth  of  C.B.  below  W.L.  x      Tons  per  inch 

Tons  displacement 


114 


STABILITY. 


Type  of  Ship. 

Minimum  gm  in  feet. 

First-class  Battleships— Modern     . 

Do.       Older  types  and  Cruisers 

Torpedo   Boat   Destroyers     .     .     . 

Torpedo   Boats 

Steamboats 

Large  Mail  and  Passenger  Steamers 

Cargo-carrying  Steamers       .     .     . 
Tugs 

5-0 
3-5 

2-0 

1-5 

1-0 

1-0  to  2-0* 

(maintained  by  water 

ballast) 

2-0 

1-5 

Very   large 

1*5  to  6  (depending  on 

sail  area) 

Shallow  Draught  Vessels  .... 
Sailing  Ships 

In  some  very  large  modern  liners  the  gm  is  greater;  e.g.  in  Aquitania 
GM  is  4  feet. 

Approximate  Formula  for  Height  of  Metacentre. 
Depth  of   Centre  of  Buoyancy   (Normand's  Formula). 
Depth  of  C.B.  below  water-line  =  J  mean  draught  + 
volume  of  displacement        .  ,        ,  ^  ,  displacement  in  tons 

1 1 7 ^1 ;  or  i  mean  draught  +  -r;; — 7-7 -• — r 

S  X  area  of  water-plane '       *>  °  36  x  tons  per  mch 

Note. — If  a  bar  keel  is  fitted,  the  mean  draught  should  be 
taken  to  the  top  of  keel. 

Alternatively,  this  depth  can  be  expressed  as  a  percentage 
of  the  mean  draught,  which  is  about  42  for  fine  ships,  44  for 
ordinary  battleships,  and  46  for  many  merchant  vessels. 

Distance    (bm)    between    Centre   of   Buoyancy   and   Mela- 

tentre. 

(greatest  beam)^ 

BM  =  -^ li -, — 

K  X  draught 

ships,    12   in    light   cruisers,    destroyers,    cargo,    and   passenger 

steamers,    and    11    in    steam    yachts.      In  new   designs   it   is 

advisable  to  take  the  value  of  K  found  in  a  similar  ship. 

Stability  at  Large  Angles  of  Heel. 

If  the  heel  be  so  large  that  the  vertical  through  b' 
(fig.  119)  no  longer  intersects  the  middle  line  at  a  fixed  point, 
the  metacentric  method  is  no  more  applicable. 

During  the  inclination  of  the  ship  the  centre  of  buoyancy 
moves  from  B  to  b',  and  b'  lies  in  a  plane  parallel  to  a  line 
joining  g  and  g' .     The  distance  bb'  can  be  found  from  the 


where  K  is  approximately  13  in  battle- 


following    expression  :  — 


bb'  = 


V  X  gg 


where  v  =  volume  of  displacement  and  v 
of   the   wedges  ; 


volume  of  either 


STABILITY.  115 

V  X  hh'      ,  .  T     ,      .      / 

BR  =  ,  where  BR  is. perpendicular  to  B  M ; 

,                                        .     „      V  X  hh'  .     „ 

and  GZ  =  BR  -  BG  .  sin  6  = BG  .  sin  0, 

whence   Atwood's    formula    for     expressing    the     moment    of 
statical  stahility  at  any  angle  Q  is 

fivxhh')       ,  .      A 

M  =  w-j — - — -  -  (bg  .  Sin  e)f 

The  moment  of  statical  surface  stability  at  any  angle  B  is 
BR  X  w,  being  the  righting  moment  obtained  on  the 
assumption  that  the  e.g.  of  the  ship  coincides  with  B.  The 
angle  of  heel  in  fig.  119  is  bmb'  =  lsl',  and  its  sine  is  equal  to 

BR      GZ 
BM  "  GM 

Dynamical  stahility  is  deftnod  to  be  the  amount  of 
mechanical  work  necessary  to  cause  a  body  to  deviate  from  its 
upright  position,  or  position  of  equilibrium. 

Dynamical  stability  is  expressed  as  a  moment  by  multi- 
plying the  sum  of  the  vertical  distances  through  which  the 
centre  of  gravity  of  the  ship  ascends  and  the  centre  of 
buoyancy  descends  (i.e.  the  vertical  separation  of  G  and  B), 
in  moving  from  the  upright  to  the  inclined  position,  by  the 
displacement. 

In  fig.  119  during  the  inclination  of  the  ship  through  the 

angle    6,   the  centre   of  gravity  has   been  moved  through   a 

vertical  height  Gii  —  go,  and  the  centre  of  buoyancy  has  been 

lowered  through  a   vertical  distance  b'i-ch,  and  the  whole 

work  to  do  this,  or  her  moment  of  dynamical  stability  for 

the  given  angle  0,  is 

=  w{(gh  -  go)  +  (r'i  -  bh)} 

=  w(b'z  -  bg)  =  w(b'r  -  BG  .  vers  0) 

/wv  (gh-{-g'h')  „, 

=  w(^— ^--    -"^ — '  -  BG  .  vers  0) ; 

whence    Moseley's    formula    for    the    moment    of    dynamical 
stability  at  any  angle  0  is 

=  ivvigh  +  g'h')  -  (w  x  bg  .  vers  0), 
where  w  is  the  density  of  water. 

The  dynamical  stability  of  a  ship  at  any  an2ple  0  is  the 
integral  of  its  statical  stability  at  the  given  angle — that  is, 
if  M  =  the  statical  stability  and  U  the  dynamical  stability,  then 

U  =  ^^1(10, 
where  d0  is  a  very  small  angle  of  heel. 

The  moment  of  dynamical  surface  stability  is  expressed  by 
multiplying  the  weight  of  the  ship,  or  displacement,  by  the 
depression  of  the  centre  of  buoyancy  during  the  inclination 
-that  is,  for  the  angle  0 

U  =  w(b'i  -  bh). 


116 


STABILITY. 


The  Curve  of  Statical  Stability  is  a  curve  used  to  record 
the  value  of  the  stability  lever  (Gz)  of  a  vessel  at  any  given 
angle  of  heel. 

Fig.  123. 
curve  of  statical  stability  of  an  ibonclad  with  high  freeboard 


Method  of  Construction. — Calculate  the  length  of  the  arm 
of  the  righting  couple,  or  GZ  (see  fig.  119),  for  several  succes- 
sive angles  of  heel  taken  between  the  upright  position  and 
that  at  which  the  leng'th  of  the  arm  becomes  zero  ;  set  the 
lengths  of  these  arms  off  as  ordinates  (see  fig.  123)  from 
a  base  line  the  abscissa)  of  which  represent  to  scale  the 
respective  angles  of  heel  :  a  curve  bent  through  the  extremi- 
ties of  these  ordinates  will  form  a  curve  of  statical  stability- 

Note.—The  length  of  the  perpendicular  at  57'3°  (one  radian) 
intercepted  between  the  tangent  at  the  initial  portion  of  the 
curve  aaid  the  base  line  is  equal  to  the  metacentric  height. 

The  Curve  of  Bynamiccl  Stability  is  constructed  in  a 
similar  manner  to  that  of  the  curve  of  statical  stability,  with 
the  exception  that  the  various  lengths  of  the  arm  (b'z  —  bg) 
=  (b'r-«g  vers  6)  (see  fig.  119)  are  taken  as  ordinate? 
instead  of  GZ.  Or  preferably  the  curve  is  obtained  by  in- 
tegrating the  statical  curve.  The  area  up  to  each  ordinate  of 
the  statical  curve  expressed  in  degrees  X  feet  is  divided  by 
67"3°  and  set  up  as  an  ordinate  of  the  dynamical  curve. 


Fig.  124. 
curve  of  dynaincaii  stability  of  an  ironclad  with  high  freeboard. 


^^'  '""^ 


M*     tt»      «•     !«•    as*     »•  35 "     «-      4S»   so"    SS«    «o'    66*     lo"    7s'    «0'    tS     »• 

BCALK  tr   •'■cncES    rta    amblb     tr  tiuk 

2sote.—T\\Q  angle  at  which  the  statical  lever  vanishes  (and 
at  which  the  dynamical  lever  is  a  maximum)  is  termed  the 
range  of  stability. 


STABILITY. 


117 


Fig.  125. 
Cross  curves  of  Stability. 


Cross   Curves   of   Stadility. 

These  curves  may  be  termed  'vertical  curves  of  stability  ' ; 
they  consist  of  curves  of  righting  levers  at  v.arious 
draughts  or  displacements  for  certain  fixed  angles  of  heel. 
They  hold  a  somewhat  similar  relation  to  the  ordinary 
curves  of  stability  as  the  body  plan  of  a  ship  does  to  its  water 
plane. 

For  cross  curves  (see  fig. 
125)  the  righting  levers  are  cal- 
culated at  certain  fixed  degrees 
of  heel  at  various  displace- 
ments, and  the  levers  are  set  up 
as  ordinates  from  an  axis  the 
abscissa}  of  which  represent  the 
displacement  at  which  the  levers 
for  the  fixed  degree  of  heel  are 
found. 

A  number  of  such  curves  are 
constructed  for  various  inclina- 
tions, and  set  off  as  in  fig.  125. 

Fig.  126. 
jo>  Curve  of  Stability  at  soo  tns.Displac  Mt.  T. 


15  30  45  60  75 

Scale  of  angle  of  heel  in  degrees. 

For  finding  such  curves  at  various  draughts  and  angles  of 
Fig.  127.  heel,  say  at   15°   (see 

A    /  fig.     127),   divide   the 

body  plan  by  a 
number  of  parallel 
planes  representing 
various  draughls  of 
water  or  displace- 
'•^-.Tients. 


^       Drop     a     perpen- 
^  dicular     through     thp 
w^  point       where        the 
highest  water-line  cuts 
the  middle  line  of  the 
ship,  and  then  calcu- 
late  (by  the  methods 
indicated  hereafter)  the  horizontal  distances  d^,  d^,  d^,  etc.,  of 


Fig.  128. 


Scale  of  feet  for  levers. 


113  STABILITY. 

the  centre  of  buoyancy  up  to  each  inclined  water-plane  from 
the  vertical  ab. 

By  assuming  the  centre  of  gravity  to  be  at  s,  and  fixed 
there  -for   all   draughts,   the  dis- 
tances f/j,  d2,  d^,  etc.,  would  be 
the    righting    levers    at    the    dis-     7oo- 
placement?,  up  to  the  respective 
water-planes  w^,  Wg,  W3,  W4.  q, 

These  lengths  are  then  set  off  5^00 
as  ordinates  along  an  axis  having  ^ 
the  several   displacements   up   to  o  ^ 
the  water-planes  as  abscissae.  ^^ 

The  actual  righting  levers  can  w 
then    be    determined,    when    the  r 
correct   positions   of   the   centres  o 
of  gravity   corresponding  to   the  s 
various    displacements    are    fixed,  ZgQo 
by  multiplying  the  respective  dis-  5 
tancej    h^,    h^,    A3,    etc.,  of  the  h 
actual  centres  of  gravity  g^^  gc,,  z  200 
.^3,  9i  below  S  by  the  sine  of  the  ' 
angle    of    heel,    and    adding    this 
length  to  the  arms  already  found 
(see  fig.   128). 

The  actual  righting  lever  for  the  displacement  up  to  W4 
would  be  equal  to  di  +  /^4  sin  15°  =  di  +  ^4,. 

Up  to  Ws,  W3  it  would  be  equal  to  dz  +  Jib  sin  15°  =  <?3  +  h,  etc. 

Should  any  of  the  centres  of  gravity  be  above  the  point  S, 
a  deduction  would  have  to  be  made  equal  to  the  distance  h  of 
the  centre  of  gravity  above  s  multiplied  by  the  sine  of  the 
angle  of  heel. 

To  Calculate  the  Statical  and  Dynamical  Stabilities  of 
A  Vessel  at  Successive  Angles  of  Heel. 

Among  the  various  methods  that  are  used  for  calculating 
the  statical  and  dynamical  levers,  three  are  here  described— « 
(a)  Barnes'  method,  (b)  the  direct  method,  (c)  the  integrator 
method  ;  the  last  named  is  by  far  the  quickest  and  most 
convenient.  Either  equidistant  sections  may  be  employed 
using  Simpson's  rule  or  specially  spaced  sections  with  Tcheby- 
cheff's  rule  (see  displacement  sheet,  p.  100).  The  former  may 
obviate  the  preparation  of  a  special  body  plan,  but  the 
latter  rule  is  generally  more  expeditious  on  the  whole.  It  is 
generally  assumed  that  all  weights  are  fixed,  all  openings  in 
the  sides  and  decks  closed  and  made  watertight,  all  appendage? 
can  be  neglected,  and  that  no  change  oi  trim  takes  place 
during  inclination. 

Barnes'  Method. 
-Prepare  a  body  plan  (fig.  130)  in  which  alj 


\V4 


1.  Body  Plan. 


the  sections  are  taken  perpendicular  to  the  load  water-line, 


Fig.  180. 


STABILITY.  119 

and  at  equal  distances  apart  (if  Tchebycheff's  method  be 
employed  the  sections  are  spaced  as  shown  in  the  dis- 
placement sheet,  p.  100).  In  constructing  it  the  sections 
should  be  made  fair  continuous  curves,  any  irregularities 
which  might  be  caused  by  embrasures,  etc.,  being  left  out 
P^fj  J09  (as  shown  in  full  lines  in 

_^  fig.  129,  where  the  dotted 

[      \  lines     show      the     actual 

«-^j'i?i'«l.  ce;.tion    of    vessel),     they 
\    J  being    treated    separately 
\/      afterwards  as  appendar/cs, 
"     When    there  are    appen- 
dages it  is  alsj  necessary 
to  have  correct  sheer  and 
half-breadth  draughts,  in 
order     to    calculate   their 
volume,  etc. 

2.  Angular  Interval. — The  body  plan  has  now  to  be  crossed 

by  a  number  of  lines, 
radiating  from  the 
middle  point  of  tho 
load  wa^.er-plane,  and 
at  equiangular  in- 
tervals from  6^  to  10°, 

2«  arranging  if  possible 
that  one  passes 
through  the  edge  of 
the  upper  continuous 
deck  amidships. 

The  equiangular 
interval  is  determined 
as  follows  : — Divide  the  angle  which  the  radiating  line,  passing 
through  the  edge  of  the  upper  deck,  makes  with  the  load 
water-line  into  such  a  number  of  equiangular  intervals  that 
the  line  passing  through  the  edge  of  the  upper  deck  becomeai 
a  stop-point  in  the  integration  to  which  these  radiating  lines 
will  be  afterwards  treated.  If  Simpson's  first  rule  is  used 
the  number  of  intervals  must  be  even  ;  if  his  second  rule, 
a  multiple  of  three  must  be  used,  and  so  on. 

3.  Measuring  the  Ordinates. — The  ordinates  of  tho 
immersed  and  emerged  sides  of  the  various  inclined  longi- 
+udinal  water-planes  are  measured  off  right  fore  and  aft  for 
each  successive  angle  of  heel  from  the  middle  line  of  the 
ship,  and  entered  upon  a  set  of  tables,  styled  preliminary 
tables,  under  their  proper  heading.  One  of  these  tables  is 
necessary  for  each  separate  angle  of  heel. 

4.  Preliminary  Table. — Three  operations  arc  performed 
(see  p.  122)  upon  the  ordinates  entered  in  these  tables.  Firstly, 
^hey  are  affected  by  a  set  of  Simpson's  multipliers,  in  order 


^MD^ 


120  STABILITY. 

to  find  a  function  for  the  area  of  the  immersed  and  emerged 
sides  of  the  respective  radial  planes.  Secondly,  the  squares 
of  the  ordinates  are  affected  hy  the  same  set  of  multipliers  in 
order  to  find  a  function  for  the  moment  of  the  immersed  and 
emerged  sides  of  the  respective  radial  planes.  Thirdly,  the 
cubes  of  the  ordinates  are  aflPected  by  the  same  set  of  multi- 
pliers in  order  to  find  a  function  for  the  moment  of  ineHia  of  the 
immersed  and  emerged  sides  of  the  various  radial  planes  about 
the  middle  line  of  ship. 

5.  Combination  Tables  (see  p.  123). — The  results  obtained 
in  the  preliminary  tables  are  made  use  of  in  these  tables  to 
determine — 

(1st)  The  area  of  the  various  inclined  water-planes,  together 
with  their  centres  of  gravity. 

(2nd)  The  volumes  of  the  assumed  wedges  of  immersion  and 
emersion. 

(3rd)  The  position  of  the  true  water-jDlanes  at  the  different 
angles  of  heel. 

(4th)  The  moments  of  the  corrected  wedges  of  immer- 
sion and  emersion. 

6.  Areas  of  the  Inclined  Water-planes. — The  area  of  an  inclined 
water-plane  is  easily  found  for  innj  angle  of  heel  by  adding 
together  the  sums  of  the  functions  of  the  ordinates  for  the 
immersed  and  emerged  sides  of  the  respective  water-planes, 
and  multiplying  the  result  by  ^  the  longitudinal  interval  if 
Simpson's  first  rule  is  used.* 

7.  Centre  of  Gravity  of  the  Inclined  Water-planes. — To  find 
the  distance  of  the  centre  of  gravity  of  any  inclined  water-plane 
relatively  to  the  middle  line  of  the  ship,  proceed  as  follows : 
— Take  the  difference  between  the  sums  of  the  functions  of  the 
squares  of  the  ordinates  for  the  immersed  and  emerged  sides  of 
the  water-plane  ;  divide  the  result  by  2  and  multiply  the 
quotient  by  ^  the  longitudinal  distance  between  the  ordinates, 
if  Simpson's  first  rule  is  used.  That  product  divided  by  the 
area  of  the  water-plane  will  give  the  distance  of  its  centre  of 
gravity  from  the  middle  line. 

8.  Volumes  of  Assumed  Wedges. — Take  the  sums  of  the  func- 
tions of  the  squares  of  the  ordinates  for  both  sides  of  each  of 
the  radial  planes  contained  in  the  wedges  of  immersion  and 
emersion,  and  enter  them  in  their  proper  column  in  the  com- 
bination table,  and  affect  them  by  a  proper  set  of  multipliers  ; 
add  their  results  together,  subtract  the  lesser  sum  from  the 
greater,  and  divide  the  result  by  2.  The  quotient  multiplied 
by  ^  the  longitudinal  distance  between  the  ordinates,  if  Simp- 
son's first  rule  is  used  (this  division  by  3  is  generally  done  in  the 
preliminary  tables) :  this  final  product  multiplied  by  ^  of  the  equi- 
angular interval  in  circular  measure,  if  Simpson's  first  rule  is  again 

*  iVip/e.— The  division  by  3  is  generally  done  in  the  preliminary  tables. 


STABILITY.  121 

tised,  will  give  the  difference  between  the  volumes  of  the  assumed 
wedges  of  immersion  and  emersion.  If  there  are  any  appendages 
the^ necessary  additions  or  deductions  are  made  here. 

9.  CWrectififf  Layer. — If  the  volume  of  the  assumed  wedge  of 
immersion  exceeds  that  of  the  wedge  of  emersion,  it  shows  that 
the  displacement  up  to  the  radial  plane  is  too  great,  and  that  to 
find  the  true  water-plane  a  parallel  layer  must  be  taken  away 
from  the  assumed  wedges ;  but  if  the  wedge  of  emersion 
exceeds  that  of  immersion,  a  parallel  layer  must  be  added  to  the 
wedges. 

The  thiclincss  of  this  layer  is  found  by  dividing  the  dif- 
ference between  the  volumes  of  the  two  assumed  wedges  by  the 
area  of  the  proper  radial  water-plane,  having  made  any  addi- 
tions or  deductions  in  the  case  of  appendages. 

10.  ^laments  of  Wedges  for  Statical  Stability. — The  sums  of 
the  functions  of  the  cubes  of  the  ordinates  for  both  the  im- 
mersed and  emerged  wedges  are  placed  in  the  proper  column  in 
the  combination  table,  and  are  affected  by  the  same  set  of 
multipliers  as  were  determined  for  the  sums  of  the  functions 
of  the  squares  ;  the  products  are  multiplied  by  the  various 
cosines  of  tlie  angles  of  inclination  made  by  the  radial  planes 
with  the  load  water-line  ;  the  products  are  then  added  together 
and  the  sum  divided  by  3  ;  the  quotient  is  then  multiplied  by  ^ 
the  angular  interval,  and  that  product  by  ^  the  longitudinal 
interval,  between  the  ordinates,  if  Simpson's  first  rule  has  been 
used  (this  division  by  3  is  generally  done  in  the  preliminary 
tables) :  the  final  result  will  be  the  moment  of  the  wedges  about 
a  line  perpendicular  to  the  radial  plane,  and  passing  through 
the  middle  point  of  the  load  water-plane.  The  corrections  for 
the  moments  of  the  appendages  must  now  be  added  or  .sub- 
tracted, as  the  case  may  be,  also  the  correction  for  the  layer,  if 
any,  must  be  done  here,  its  moment  being  found  l)y  multi- 
plying its  volume  by  tlie  distance  of  the  centre  of  gravity  of  its 
water  plane  from  the  middle  point  of  the  load  water-plane.  If 
the  centre  of  gravity  of  the  layer  lies  towards  that  side  for 
which  the  assumed  wedge  is  the  greater,  the  correction  must  be 
deducted  ;  if  it  lies  towards  the  opposite  side,  it  must  be  added. 
This  final  result,  being  divided  by  the  total  volume  of  displace- 
ment, will  give  the  length  of  the  arm  BR  (see  fig.  119).  Multiply 
the  heiglit  of  the  centre  of  gravity  above  the  centre  of  buoyancy 
by  the  sine  of  the  angle  of  heel,  and  subtract  the  product  from 
Bii;  the  remainder  will  be  the  length  of  the  arm  of  the  righting 
couple  GZ  ;  gz  multiplied  by  the  displacement  in  tons  will  give 
the  righting  moment,  or  statical  stability,  of  the  ship  for  the 
given  angle  of  heel. 

11.  Moments  of  the  Wedges  for  DynamicalStahility. — This  result 
is  determined  in  a  manner  somewhat  similar  to  that  pursued 
for  the   statical  stability,  the  only  difference  being  that  the 


122 


PRELIMINARY    TABLE    FOR    STABILITY. 


Preliminary  Table  eor  Stability  at  30°  Angle  of  Heel. 

1 

Ordi- 
nates 

23 

.2 

fS     o 

Squares 

of 
Ordi- 
uates 

1 

Functions 

of 
Squares 

1 

Cubes 

of 
Ordi- 
uates 

i 

1 

Functions 

of 

Cubes 

Immersed  Wedge.                                  | 

1 

•8 

i 

•4 

•6 

i 

•3 

•5 

h 

•3 

U 

8-1 

2 

16-2 

65-6 

2 

131-2 

531-4 

2 

1062-8 

2" 

14-2 

1 

14-2 

201-6 

1 

201-6 

2863-3 

1 

2863-3 

2^ 

17-8 

2 

3o-6 

316-8 

2 

633-6 

5639-7 

2 

11279-4 

3 

20-5 

u 

30-7 

420-2 

H 

630-3 

8615-1 

H 

12922-7 

4 

20-4 

4 

81-6 

416-2 

4 

1664-8 

8489-7 

4 

33958-8 

5 

20-2 

2 

40-4 

408-0 

2 

816-0 

8242-2 

2 

16484-4 

6 

20-2 

4 

80-8 

408-0 

4 

1632-0 

8242-2 

4 

32969-6 

7 

20-2 

2 

40-4 

408-0 

2 

816-0 

8242-2 

2 

16484-4 

8 

20-2 

4 

80-8 

408-0 

4 

1632-0 

8242-2 

4 

32969-6 

9 

20-2 

U 

30-3 

408-0 

U 

612-0 

8242-2 

u 

12363-6 

n 

20-3 

2 

40-6 

412-0 

2 

824-0 

8363-6 

2" 

16727-2 

10 

18-8 

1 

18-6 

353-4 

1 

353-4 

6644-7 

I 

6644-7 

10^ 

15-8 

2 

31-6 

249-6 

2 

499-2 

3944-3 

2 

7888-6 

11 

10-6 

h 

5-3 

112-4 

h 

56-2 
3)10502-6 

1191-0 

i 

595-5 

3)547-3 

Immersed 

3)204972-9 

182-4 

3500-9 

.    68324-3 

Emerged 
Both  wedg 

.    58590-4 

es  126914-7 

Emerge 

D  Wedge. 

1" 

M 

^ 

-5 

1-2 

h 

•6 

1-3 

h 

•  7 

1^ 

6-5 

2 

13-0 

42-2 

2 

84-4 

274-6 

2 

549-2 

2 

10-9 

10-9 

118-8 

1 

118-8 

1295-0 

1 

1295-0 

n 

141 

28-2 

198-8 

2 

397-6 

2803-2 

2 

5606-4 

3 

16-9 

U 

25-3 

285-6 

u 

428-4 

4826-8 

u 

7240-2 

4 

20-0 

80-0 

400-0 

4 

1600-0 

8000-8 

4 

32003-2 

5 

21-2 

42-4 

449-4 

2 

898-8 

9528-1 

2 

19056-2 

6 

21-5 

4 

86-0, 

462-2 

4 

1848-8 

9938-4 

4 

39753-6 

7 

21-2 

„ 

42-4 

449-4 

2 

898-8 

9528-1 

2 

19056-2 

8 

20-1 

80-4 

4040 

4 

1616-0 

8120-6 

4 

32482-4 

9 

17-5 

n 

26-2 

306-2  1 

H\ 

459-3 

5359-4 

H 

8039-1 

91 

15-4 

2 

30-8 

237-1 

2 

474-2 

3652-3 

2 

7304-6 

10 

12-5 

1 

12-5 

156-2 

1 

156-2 

1953-1 

1 

1953-1 

10| 

8-9 

2 

17-8 

79-2 

2 

158-4 

705-0 

2 

1410-0 

11 

3-5 

i 

1-7 

12-2 

ij 

6-1 

42-8 

^ 

2L-4 

3)508-1 

3)9146-4 

3)175771-3 

169-3 

3048-8 

58590-4 

COMBINATION    TABLE    FOR    STABILITY. 


123 


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124  STABIUTY. 

sums  of  the  functions  of  the  cubes  are  multiplied  by  the  sines 
of  the  various  angles  of  inclination  instead  of  the  cosines;  the 
sum  of  the  products  so  obtained  being  divided  and  multiplied 
by  the  same  numbers  as  were  used  for  the  statical  stability,  in 
order  to  find  the  moment  of  the  wedges  uncorrected  relatively 
to  the  respective  radial  planes.  The  corrections  for  the 
appendages  are  then  made,  that  for  the  correcting  layer 
being  subtracted  in  all  cases.  The  moment  for  the  correcting 
layer  is  found  bj^  multiplying  its  volume  by  half  its  thickness, 
that  being  about  the  vertical  height  of  its  centre  of  gravity 
from  its  radial  plane.  This  final  result  divided  by  the  total 
volume  of  displacement  will  give  the  length  of  the  arm  b'  r, 
from  which  if  BG  .  vers  6  be  deducted,  the  remainder  will  equal 
the  length  of  the  arm  for  the  dynamical  stability,  or  the  vertical 
height  through  which  the  centre  of  gravity  of  the  ship  has  been 
lifted  and  the  centre  of  buoyancy  depressed. 

12.  Geometi'ical  Mode  of  Calculating  Dynauiieal  Stability. — 
The  dynamical  stability  of  a  vessel  at  any  given  angle  of  heel 
is  the  sum  of  the  moments  of  the  statical  stability  taken  at 
indefinitely  small  equiangular  intervals  up  to  tlie  given  angle 
of  heel,  and  is  therefore  equal  to  the  area  of  the  curve  of  sta- 
tical stability  included  between  the  origin  of  the  curve  and  the 
angle  in  question.  It  must  be  noticed  that  the  abscissie  of  a 
curve  of  statical  stability  is  given  in  angles,  and  therefore  the 
longitudinal  interval  is  taken  in  circular  measure. 

But,  as  the  lengths  of  the  arms  for  statical  stability  are 
generally  used  to  construct  a  curve  instead  of  the  moments  of 
stability,  the  area,  as  above  found  by  the  rule  from  such  a  curve, 
will  necessarily  give  the  length  of  the  a7'm  for  dynamical 
stability  and  not  the  moment. 

Example  (see  fig.  123). — To  find  the  length  of  the  arm 
for  dynamical  stability  at  an  angle  of  30°  inclination. 


Angles  of  Heel 


0  de2:rees 


10 
15 
20 
25 
30 


Lengths  of  Statical 
Levers  GZ 


•0 
•2 
•42 
•68 
•97 
1-30 
1-66 


Simpson's 
Multipliers 


Products 


•0 

•8 

•81 

2-72 

M)4 

5-20 

1-66 

l346 


^  of  angular  interval  in  circular  measure  =  '0291 

1316 
11844 
2632 
Dynamical  lever  for  30°  =  -382956 


STABILITY. 


125 


13.  Curre  of  Stability  for  Light  Draught. — The  lengths  o.f 
the  arms  for  this  curve  can  readily  be  approximated  from  the 
results  obtalijed  for  the  curve  in  the  load  condition. 

In  fig.  131  WL  13  the 
load  water-line,  and  wl 
the  light  water-line,  for 
the  upright  position  of 
the  vessel.  If  the  vessel 
is  inclined  through  an 
angle  0,  and  w'l'  is  the 
true  position  of  the  in- 
clined water-plane  for 
the  load  condition,  then 
the  true  position  of  the 
wafer-plane  for  the  light 
condition  will  run 
parallel  to  it,  as  w'V.  To  determine  its  perpendicular  dis- 
tance from  w'l',  divide  the  volume  of  the  layer  contained 
between  the  light  and  load  water-planes  by  the  area  of  the 
assumed  inclined  water-plane  hh',  which  was  found  for  the 
inclined  load  condition.  Let  B  be  the  centre  of  buoyancy  for 
the  upright  load  condition,  b'  for  the  inclined  load  condition, 
and  b2  for  the  inclined  light  condition,  br  is  perpendicular 
to  the  vertical  b'm,  and  br'  is  perpendicular  to  the  vertical 
b2m'. 

Let  V  equal  volume  of  light  displacement. 
„     V      ==     volume    of    displacement     contained    between 

the  light  and  load  water-planes. 
„  c  =  distance  of  centre  of  gravity  of  assumed  in- 
clined water-plane  from  the  vertical  through 
A,  assumed  positive  on  the  emerged  side. 
„  t32  and  G'z'  =  the  lengths  of  the  arms  of  the  righting 
couples  for  the  load  and  light  condition 
respectively. 

rrn-  ,    ,       ■  f     .      fl    ,    '^(BR  -  BA  sin  0  -^  c) 

Then  g  z'  =  GZ  -  GG  sin  e  + '- 

Example.— In.  the  ship  illustrated  in  the  tables  (pp.  122, 
123)  find  the  lever  of  statical  stability  at  30°  when  light, 
assuming  the  displacement  diminished  by  200  tons,  and  the 
e.g.  raised  by  15  ft,    B  is  6*5  ft.  below  original  upright  w.L. 

Here  v  =  200  X  35  =  7000  ;  V  =  86767  -  7000  =  79800  approx. 

GZ  =  l-65;  gg'  =  1-5;  sin  6  =  §  ;  BR  =  4-78  ;  BA  =  6-5  ; 
C  =  -  1-16. 

7000(4-78  -  3-25  -  1-16) 


GZ' 


1-65-  -75  + 


79800 


•93'. 


Direct  Method. 
Lay  a  piece  of  tracing  paper  over  the  body  plan,  and  on 
it  draw  a  trial  water-line  at  the  correct  inclination.    Trace  the 


u^ 


STABILITY. 


wedge  sections,  replacing  the  curved  portions  by  one  or  two 
straight  lines  approximating  ^as  closely  as  possible  to  the 
curves.  Find,  graphically  the  areas  of  the  triangles  and 
quadrilaterals,  and  thence  determine  the  volume  of  each  wedge. 
If  these  are  not  nearly  equal  raise  or  lower  the  water-line,  and 
proceed  as  before  until  there  is  practical  equality  in  volume. 
Find  the  e.g.  of  each  triangle  or  quadrilateral  (see  p.  59) 
and  calculate  the  moment  of  its  area  about  anj'  line  perpen^ 
dicular  to  the  inclined  water-line.  Thence  find  the  moment? 
of  the  volumes  and  add.  The  total  moment  divided  by  the 
volume  of  displacement  is  equal  to  be  (fig.  119),  whence  GZ 
is  at  once  determined. 

The  direct  method  is,  perhaps,  the  most  convenient  one 
when  an  integrator  is  not  available 

Amsler-Laffon's  Mechanical  Integrator. 

By  means  of  this  instrument,  the  area,  moment,  and  moment 
of  inertia  about  any  axis,  can  be  obtained  for  any  curvilinear 
area  by  tracing  its  outline  with  a  pointer. 

Its  principal  use  is  that  of  obtaining  the  stability  of  a  vessel 
at  various  angles  of  heel  and  at  various  drafts.  It  is  usual, 
when  using  this  instrument,  to  first  calculate  the  righting  lever 
for  a  number  of  displacements  atone  inclination,  say  15°.  Then 
the  same  for  30°,  45°,  and  soon ;  the  cross  curves  being  constructed 

before  the  ordinary  curves. 

Let  fig.  132  be  a  body 
plan  drawn  for  both  sides  of 
a  ship  ;  let  wl  be  its  upright 
waterline  intersecting  the 
middle  line  at  s.  Through 
S  draw  inclined  waterlines 
at  the  required  inclinations, 
and  let  w'l'  be  any  one  of 
them,  say  at  15°.  The  first 
step  is  to  find  the  displace- 
ment at  w'  l'  as  it  is  gene- 
rally different  from  that  at 
WL.  The  pointer  is  passed  (i)  round  the  two  end  sections, 
(ii)  round  the  dividing  sections,  and  (iii)  round  the  intermediate 
sections*  ;  the  pointer  in  each  case  passing  along  the  waterline 
and  round  the  section,  as  w'l' aw'.  Readings  are  taken  at  the 
start  and  after  passing  round  (i),  (ii),  and  (iii),  so  that  after 
subtracting,  the  readings  due  to  each  of  the  three  series  of 
sections  are  known.  Reading  (ii)  is  multiplied  by  2,  and  (iii)  by 
4,  and  the  tw^o  products  added  to  reading  (i).  The  total  is  then 
multiplied  by  the  common  interval  and  the  constant  of  the 
instrument  and  divided  by  3  times  the  square  of  the  scale  used. 
The  result  is  the  volume  of  displacement,  w^hich  is  then  reduced 
to  tons. 

*  See  Simpson's  Rules. 


STABILITY. 


127 


If  in  the  same  way  ST,  the  line  through  s,  perpendicular  to 
w'l'  is  made  the  axis  for  moments,  and  the  readings  for 
moments  are  treated  in  the  same  way  as  those  for  areas,  it  is 
evident  that  tlie  final  result  will  be  the  moment  of  the  under- 
water portion  about  ST  as  axis  (obviously,  the  total  must  now 
be  divided  by  8  times  the  cube  of  the  scale  instead  of  the  square). 
This  divided  by  the  volume  of  displacement  will  give  the  per- 
pendicular distance  of  the  inclined  centre  of  buoyancy  from 
ST  ;  that  is  sz,  when  b'z  is  parallel  to  ST. 

The  righting  lever,  or  GZ,  is  equal  to  sz  +  SG  sin  Q  when  G  is 
below  s  as  at  g,  ;  and  equal  to  sz  — SG  sin  d  when  G  is  above  S. 

The  righting  lever  gz  is  set  off  at  its  proper  displacement  on 
the  cross  curve  for  15°.  This  is  done  at  different  waterlines  and 
the  cross  curve  thus  completed. 

The  following  is  the  actual  form  of  the  calculation  for  sz. 
Sections  10*6  apart.     Scale  of  body  ^"  to  1  foot. 

20 
1000' 

40 
1000' 


Machine  constants 


Areas 


Moments 


Angle  of  Hekl  15^^ 


Sections 


Initial 

End  ordi nates 
Dividing  ordinat. 
Intermediate  „   . 


Areas 


4029 
4111 
10502 
17309 


6391 
6807 


,"2 

fi 


82 
12782 
27228 


40092 


Moments 


982 


1398 
1819 


4 
412 
421 


4 
824 
1684 


2512 


Displacement  in  tons 

*  The  4  multiplier  is  the  reciprocal  of  the  scale  of  the  drawing. 


123  STABILITY. 

Tchebyoheff's  rule  (see  p.  43)  can  be  very  usefully  cm- 
ployed  instead  of  Simpson's  rule  in  the  above  ;  the  saving  of 
time  due  to  its  adoption  is,  for  a  complete  set  of  cross  curves, 
more  than  suflScient  to  compensate  for  the  time  of  preparing 
the  special  body  plan,  which  need  only  be  drawn  in  fairly 
roughly.  In  this  labour  may  be  avoided  by  using  the  sections 
numbered  2,  5,  7,  10,.  12,  15,  17,  and  20  from  an  ordinary 
body  plan  whose  equidistant  sections  are  numbered  1  to  21. 
It  will  be  found  that  these  coincide  nearly  in  position  with 
those  required  with  Tchebycheff's  rule  for  4  ordi nates,  repeated. 
This  was  pointed  out  at  Inst.  N.A.,  1915,  by  Mr.  W.  J.  Luke, 

Example. — Length  of  ship,  210  feet ;  number  of  sections,  8  ; 
scale  of  body,  J"  to  1  foot.     Machine  constants  as  before. 

20  ''lO      1 

Displacement  in  tons  =  — r-  x  16  x  ~  x  —  x  area  reading. 
J.UUU  o        oo 

4  =  the  scale 
Moment  reading  ^^  ^     2  =  ratio    of 
Area  reading  machine  co  i- 

stints. 
Note. — The  above  or  '  all-round '  method  is  the  simplest, 
since  it  gives  directly  the  stability  lever  desired.  A  more 
accurate  and  expeditious  method,  however,  is  that  known  as 
the  *  figure-eight '.  The  pointer  is  passed  around  the  outline 
of  the  wedge  sections,  only,  taking  them  in  the  opposite 
directions  on  the  two  sides  of  the  ship  ;  e.g.  commencing  at  s 
(fig.  132)  the  pointer  reaches  the  points  l',  l,  s,  w,  w',  s 
in  the  order  named.  The  result  is  to  give  the  difference  of 
the  wedge  volumes  (by  the  area  reading)  and  the  sum  of 
their  moments  (by  the  moment  reading).  If  v  be  the  original 
volume  of  displacement,  v  the  increased  volume  registered  by 
the  machine,  and  M  the  moment  registered  (the  last  two  being 
found  from  the  readings  as  in  the  '  all-round '  method), 
and  Bs  the  distance  of  the  upright  C.B.  below  s, 
M  -  V  .  BSsinS 

sz  = TT^— 

Formulae  for  Stability  Levers  in  Special  Cases. 

1.  Ship  with  concentric  circular  sections,  cylinder. — The 
metacentric  method  is  here  applicable  to  all  angles  of  heel 
and  statical  lever  GZ  =  gm  sin  d,  dynamical  lever  =  gm  vers  Q. 

2.  Wholly  immersed  vessel. — The  metacentre  and  centre  of 
buoyancy  are  coincident,  and  the  above  formulae  apply  if  B 
be  substituted  for  m. 

3.  Wall-sided  vessel,  parabolic  cylinder. — Statical  lever 
GZ  =  sin  B  (gm  +  i  bm  tan^  0). 

Dynamical  lever  =  GM  (1  -  cos  0)  +  i  BM  (sec  0  +  cos  0  -  2). 
For  BM.  its  value  when  ship  is  upright  is  intended. 


STABILITY.  129 

Change   of  Metacentrio   Height   dub   to   Small   Chang  1:9 
IN  Dimensions. 

Let  the  beam  of  a  sliip  bo  increased  by  —   of  itself,  all 

transverse  ordinates  being  augmented  in  the  same  proportion. 

Similarly  let  the  draught  be  increased  by    —  of  itself.     If 

these  changes  are  moderate,  and  the  height  of  the  e.g.  above 
the  keel  be  assumed  to  vary  as  the  draught,  the  increase  of 
metacentric   height   is   given    by — 

where  m  is  the  original  GM,  S/«  the  increase  of  gm^  and  a 
is  bg. 

If  the  beam  only  be  increased,  — =  0,  and 
?"' =-"(!+-"-)...   (2) 

111         Hi  \  1)U  ' 

If  the  draught  be  diminished  so  as  to  maintain  the  same 
displacement  as  before,  —  =  —  —    and 


-?^=i(3  +  i?)...(3) 


In  the  preceding  case  if  the  total  depth  be  unaltered  (the 
freeboard  being  increased  to  compensate  for  the  diminution 
of  draught),  and  if  /i  represent  the  height  of  the  e.g. 
above  the  keel,  originally, 


VI       n  V  111     /  ^ 


If  in  the  preceding  case  it  be  assumed  alternatively  that 
the  freeboard  is  unaltered  (the  height  of  e.g.  above  keel 
varying  as  the  total  depth  as  before),  and  if  s  represent  tha 
original  ratio  —  freeboard  -=-  draught, 

—  =  -      3  +       (4(1-  ---)  y    .  .  .    5 
m        n  y  in  ^         1  +  sO 

In  the  general  case,  determine  the  effect  on  GM  of  increasing 
the  beam  by  one  foot,  assuming  that  bm  oc  lb^/A-  The  increase 
of  GM  roughly  varies  as  that  of  beam. 

Example  1. — In  making  a  preliminary  estimate  of  the 
iimensions  of  a  new  design,  the  beam  is  assumed  36  feet,  the 
Jistance  bo  is  8  feet  and  GM  is  1  foot.  It  is  desired  to  double 
the  metacentric  height,  while  maintaining  the  draught  un* 
altered.    Find  the  beam  required. 

Using  formula  (2),  Zm  =  1,  m  -^  1,  a  —  Q, 

Whence  -(1  +  8)  =  1,  or  —  =  i. 
Ill  '  n     18 


180  STABILITY. 

Therefore  the  beam  required  is  36  (1  +  ^)  or  38  feet. 

Note  that  if  it  is  desired  not  to  alter  the  displacement,  the 
ength  must  be  diminished  by  ^  of  itself. 

Example  2. — In  a  battleship  having  beam  89  feet,  mean 
draught  27  feet,  GM  5  feet,  a  above  water-line  6J  feet,  and 
BG  18  feet,  find  the  effect  on  the  metacentric  height  of  in- 
creasing the  beam  by  1  foot,  assuming  that  owing  to  a  change 
in  the  distribution  of  weights  the  e.g.  is  0*35  feet  higher 
above  the  water-line  in  the  new  design.  The  displacement 
and  length  are  assumed  unaltered. 

Using  formula  (3),  m  =  6,  ni  =  ■^,  a  =  18. 

Whence  -y  =  ^(8-t-~|)  =  -195  or  5w  =  -98  feet. 

But  this  assumes  that  the  height  of  G  above  water-line 

becomes  6-5  (l  +  — )  or  6-5  (l  -  ^7.)  or  6-43  feet;  it  is  actually 
6 .83  feet.       ^        ">^  ^        ^^' 

Hence  the  metacentric  height  is  5  +  '98  —  (6-85  —  6-43)  = 
5 -56  feet. 

Alteration  of  Stability  Curve  due  to  Small  Changes  in 
Dimensions. 

Assume  the  beam  increased  by  —  of  itself,  and  the  draught 

1  '^1 

by  —  of  itself  as  above.     This  process  is  applicable  to  any 

two  ships  of  fairly  similar  form,  even  if  they  depart  somewhat 
from  exact  proportionality.  Given  the  curve  of  statical 
stability  (Gz)  for  the  first  ship,  it  is  required  to  construct 
the  corresponding  curve  for  the  desired  vessel  without  con- 
structing the  body  plan  or  performing  the  usual  calculations. 
EuLE. — I.  Construct  the  curve  of  dynamical  stability  of 
the  first  ship  by  taking  areas  of  the  GZ  curve  (see  p.  116). 
Two  or  three  spots  are  sufficient,  as  great  accuracy  is  not 
required. 

2.  Corresponding  to  the  angle  B  at  which  the  stability  lever 
is  required  in  the  new  ship,  determine  an  angle  <p  from  the 
formula-  (i  +  i)  tan^  =  (l  +  i)  tanS. 

3.  Determine  GZ,  the  statical  lever,  and  z  the  dynamical 
stability  lever  for  the  original  ship  at  the  angle  <p. 

4.  Determine  Zm,  th©  increase  of  metacentric  height,  by 
the  methods  of  the  previous  page. 

5.  Then  the  stability  lever  g'z'  of  the  new  ship  at  the 
angle  0  is  given  by — 


g'z'  -  gz  =  5m  sin  0  +  i  (—  H — )  (gz  —  m  sin  <p) 

+  if )  (gz  cos  2  4)  +  2  sin  2  (?  (a  +  z)  -  (3??i  +  4a)  sin  (p). 

\  7li       111' 


STABILITY. 


131 


By  calculating  g'z'  for  about  3  values  of  0,  the  stability 
curve  can  be  described  by  the  aid  of  the  tangent  at  the  origiu 
as  given  by  the  gm. 


Longitudinal  Stability. 

Definitions. — 1.  Tlie  centre  of  flotation  is  the  centre  of 
gravity  of  the  water-plane  ;  it  is  denoted  by  F  in  %.  133. 
For  longitudinal  inclinations  without  change  of  displacement 
the  water-planes  intersect  on  a  transverse  axis  passing  through 
the  centre  of  flotation. 

Pig.  133. 


^ 


2.  The  difference  between  the  draught  forward  and  that 
feft  is  termed  the  trim.  If  the  former  is  greater  the  trim 
Is  by  the  bow,  and  vice- versa.  When  not  stated  the  draughts 
pre  supposed  taken  at  the  perpendiculars  ;    they  are  actually 

neasured  at  the  draught  marks  which  are  frequently  placed 
the  extremities  of  the  straight  keel. 

3.  Change  of  trim  is  the  sum  of  the  changes  of  draught 
forward  and  aft  if  one  is  increased  and  the  other  diminished  ; 
>therwi8e  it  is  the  difference  between  the  changes  of  draught. 

To  determine  the  draughts  and  trim  at,  the  draught  marks 
jiven  those  at  the  perpendicular,  and  the  converse. 
Let  L  =  length  of  ship  between  perpendiculars. 
a,  b  =  distance  of  forward  and  after  draught  marks  from 
amidships. 
Di,  D.2  =  draughts  at  F.P.  and  A.P. 
D3,  D4  =  draughts  at  forward  and  after  draught  marks. 

+  -  (Di  -  Da). 


D3  = 


D4  = 


2 
Di  +  Da 


+  |(-> 


Dj. 


132  STABILITY. 

a  +  b  ,  , 

D4  -  I>8  =   — Z (D2  -  Di). 

J-l 

(a  +  &)  Di  =  Da  (  2  +  &  )  — D4  (  I  -  «  ) 

{a  +  b)i)2  =  D4{^+  a,)-Ds(^~  -  b) 

To  determine  the  displacement  of  a  vessel  floating  out  of 
her  designed  trim. 

Let  D   bo  mean  draught  amidships,   w  the  corresponding 

displacement  as  obtained  from  the  displacement  slieet,  T  the 

tons  per  inch,  d  the  number  of  inches  excess  of  trim  by  tha 

stern,   L   the   length   in   feet   between   perpendiculars,   and   c 

the  distance  of  the  centre  of  flotation  abaft  amidships  in  feet. 

cd" 
Then  virtual  mean  draught  is  D  -\ — — 

L 

Hence  the  displacement  is        W  H tons. 

Li 

Ex. — In  a   ship   where   l  =  400,   c  ^  15,   t  =  80,   the  dis- 
placement  deduced    from   the    mean   draught   is    14,000    tons 
where  the  ship  has  a  trim  of  2  feet  from  the  bow.     If  the 
normal  trim  bef  1  (Foot  by  the  stern,  find  the  true  displacement. 
Here  d  =  —  36",  and  increase  of  displacement  is 
80X15X36  ,^^, 
400 =-108  tons. 

Hence  displacement  is  14,000  -  108  -=  13,892  tons. 

Note. — 1.  The  distance  c  expressed  as  a  fraction  of  the  ship's 
length  has  the  following  average  values  : — Battleship  ^,  light 
cruiser  ^,  t.b.  destroyer  ^,  steam  yacht  -^,  channel  steamer,  ■^, 
cargo  steamer  1^. 

2.  The  centres  of  buoyancy  and  gravity  lie  abaft  the  midship 
section  at  a  distance,  which,  expressed  as  a  fraction  of  the 
ship's  length,  has  the  following  average  values  : — Battleship  ^, 
light  cruiser  -gV.  t.b.  destroyer  ^,  steam  yacht  1^,  channel 
steamer  ^,  cargo  steamer  0. 

3.  For  a  change  of  trim  t  without  change  of  displacement,  the 

t      tc  t       tc 

draught  forward  is  altered  by—  +  —  and  that  aft  by  7,  —  — 

J         L  ^         li 

To  -find  the  changes  of  draught  and  trim  in  passing  from 
salt  to  fresh  water^  and  vice  versa. 

Let  the  symbols  w,  T,  and  D  above  refer  to  salt  water.  Let 
8  inches  be  the  sinkage  in  fresh  water,  and  d'  the  final  meap 
draught. 

•  9w        w 
Then  D'  =  D  + 8/12;    i=^^  =  ^:^^ 

It  is  assumed  above  that  the  fresh  water  occupies  35*9  cubid 


STABILITY.  133 

feet   to    the   ton.      If   the   water    is    brackish,    and    occupies 
35  +  a:  cubic  feet  to  the  ton,  the  latter  formula  becomes 

3oT 

The  change  of  trim  is  usually  very  small.     If  c'  be  the 

distance  of  the  centre  of  flotation  abaft  the  centre  of  buoyancy, 

and  M  the  moment  to  change  trim  in  salt  water,  the  change 

of  trim  by  the  bow  on  pas.-ing  from  salt  to  brackish  water  is,  in 

X  c'w  c  w 

inches,  ;  or  ^^^t"^  for  fresh  water  where  a;  is  -9. 

Ex. — Find  the  changes  of  draught  and  trim  in  a  light 
cruiser  on  passing  into  fresh  water  if  w  =  3000,  t  =  25, 
M  =  650,  c'  =11. 

3000 
Increase  of  mean  draught  =  ■  ^  os^or:  "^  3*1  inches. 

r^x,  *^   .       ,       .,       ,.  11X3000         1    o  •      1 

Change  of  trim  by  the  bow  =  ——: — ^^^=  1-3  inches. 

To  determine  the  positions  at  which  a  weight  must  be 
added  or  removed  so  as  to  leave  the  draught  at  one  end 
constant. 

EULE. — To  maintain  constant  draught  at  a  distance  y  abaft 
(or  before)  the  centre  of  flotation,  place  the  weight  at  a  distance 

X  before  {or  abaft)  the  centre  of  flotation,  where  a;  =  — .     If 

constant  draught  is  desired  at  either  perpendicular,  the  two  points 

2  M 
for  the  weight  are  situated  at  a  distance very  nearly  from  the 

C.F.     This  distance  is  about  ,  or  about  ry:  in  many  ships. 

To  determine  the  vertical  height  of  the  longitudinal  meia- 
"sentre  above  the  centre  of  buoyancy. 

Divide  the  moment  of  inertia  of  the  water-plane  relatively 
;o  a  transverse  axis  passing  through  the  centre  of  flotation  by 
;he  volume  of  displacement  (for  example,  see  displucemcni; 
»heet  and  explanation  on  pp.  94,  99). 

Note. — This  height  is  frequently  greater  than  the  ship's 
'     ■•th,  so   that   i!G   is  negligible  in  comparison;    then  Gil  = 

.i[)proximately. 

Moment  to  alter  trim  of  a  vessel. — In  fig.  133  let  the 
'^  weight  P  be  moved  forward  through  a  longitudinal  distance 
I,  changing  the  water-line  from  wl  to  w'l'. 

r_,,  ,  W.GM    W  X  GM  X  trim  in  feet     , 

Then  PcZ  =  w.GGi  =  — T~^ 5   hence  trim 

.     ^          12PfZL  ^ 

n  inches  = 


134  STABILITY. 

The  product  vd  is  the  moment  causing  trim  ;  if  several 
weights  are  moved,  their  moments  are  added,  allowing  for 
sign.      Note    that    the    moment    to    change    trim   one   inch    is 

equal  to   the   expression   — zr^r- — .       This    is    fairly    constant 

La  Li 

within  moderate  changes  of  draught,  and  practically  unaffected 
by  vertical  shifts  of  weight. 

Ap])roa.-imate    ■formiilce.  2 

1.  (J.  A.  Normand).     Long,  bm  =  13,000  t",  L  =  length  on 

L.W.L  in  feet,  B  =  beam  in  feet,  V  =  volmne  of  displacement  in 
cubic  feet,  T  =  tons  per  inch. 

2.  (Derived  from  the  preceding)  ^2 

Moment  to  change  trim  1"  =  30  — 

B 

3.  Moment  to  change  trim  1"  =  l^b/10,000  in  ships  ol 
ordinary  form,  TL  -^  18-5. 

Effect  of  Adding  Weights  of  Moderate  Amount. 

The  weights  added  are  supposed  insuflBcient  to  affect 
appreciably  the  transverse  stability,  or  to  cause  relatively 
large  heel,  trim,  or  immersion. 

EuLE. — Find  the  distance  of  each  weight  from  the  middle 
line  plane  and  from  amidships.  Calculate  the  moments, 
positive  and  negative  (weights  removed  are  considered  nega- 
tive), and  add 

Mean  sinkage  =  weight  added  -r  tons  per  inch. 

„    ,  .     ,  „      transverse  moment 

Heel  m  degrees  =  57-3  X  tt—^j : r„ 

^  displacement  x  gm 

Trim  in  inches  = 

longitudinal  moment  about  centre  of  flotation 

moment  to  change  trim  1  inch. 

__  longitudinal  moment  about  ^Hh  yo  X  c.F.  abaft  ^j 

moment  to  change  trim  1  inch 
using  +  sign  when   the   net  weight  [iv]  added  or  subtracted  is 
before  amidships. 

EiTECT  OF  Adding  Weights  of  Considerable  Amount. 

Eule. — Add  the  weights  and  their  moments  as  above,  in- 
cluding in  addition  the  vertical  moments  required  to  find  the 
rise  or  fall  of  the  e.g. 

The  new  mean  draught  is  found  from  the  curve  of  dis- 
placement, or  more  accurately  from  the  curve  of  tons  per  inch, 
by  estimating  the  mean  tons  per  inch  between  the  twq  water- 
lines.  If  necessary  make  the  correction  due  to  the  positiot 
of  the  centre  of  flotation  as  described  on  p.  132. 

To  obtain  the  heel  find  first  the  vertical  position  of  G ;  froir 
the  metacentric  diasvam  the  new  GM  is  obtained.     The  latera 


STABILITY.  135 

movement  (gg')  of  G  is  found  by  dividing  the  transverse  moment 
by  the  new  displacement.  A  moderate  angle  {0)  of  heel  is  given 
by  the  formula  tan  6  =  gg'/gm.  If  6  is  very  large,  construct 
a  curve  of  stability  for  the  new  condition,  using  the  cross  curves, 
and  find  by  trial  the  angle  9  at  which  the  relation  Gg'  =  GZ  sec  6 
holds. 

For  the  trim  tho  method  given  on  the  preceding  page  is 
usually  sufficiently  accurate.  If,  however,  the  Binkage  is  very 
great,  construct  a  curve  of  moment  to  change  trim  1  inch  on 
a  base  of  draught,  also  one  giving  the  longitudinal  position 
of  the  centre  of  buoyancy.  Then  at  the  original  displacement 
if  the  trim  be  by  the  stem,  the  distance  of  0  abaft  b  is  equal 
to  the  trim  in  inches  X  moment  to  change  trim  1  inch  at 
that  draught  (found  from  the  curve)  -f  displacement. 
Knowing  the  longitudinal  position  of  B  from  the  curve,  that 
of  G  13  obtained.  The  change  in  this  due  to  the  added  weights 
is  then  determined  ;  and  the  above  process,  reversed  and 
using  the  final  moment,  positions  of  B  and  G,  and  displace- 
ment, gives  the  final  trim. 

Examples  of  the  above  methods  are  given  in  the  inclining 
experiment   described   below. 

To    Dl'lLRMINE    THE    VERTICAL    POSITION    OF    A    ShIP'S    CeNTRE 

OF  Gravity  by  Experiment. 

In  fig.  134  let  mz  be 
the  upright  axis  of  a  ship  ; 
her  centre  of  gravity  then 
lies  somewhere  in  that  axis. 
M  is  the  metacentre,  and 
GAI  its  vertical  height  above 
the  centre  of  gravity  G. 

If  a  weight  P  be  moved 
transversely  through  a  dis- 
tance PP'  =  d,  it  will  heel 
the  vessel  over  through  an 
angle  6,  and  her  centre  of 
gravity  will  then  shift  in 
a  direction  GO'  parallel  to 
that  in  which  the  centre  of  gravity  of  the  weight  has  been 
Bhifted.  Let  mt  be  parallel  to  gg'  and  tg'  parallel  to  gm  ;  let 
P  =  weight  shifted  in  tons,  and  W  =  displacement  of  ship  in 
tons  :   then 


MT  =  GG 


V  y  d 


,  and  GM  =  gg'  cotan  9 


V  X  d 


cotan  9. 


w    '  W 

In  practice  the  ballast  is  usually  in  the  form  of  pig  iron 
arranged  in  two  parallel  rows  on  the  port  and  starboard  sides 
■''©f  the  upper  deck. 

The  method  of  performing  the  experiment  is  illustrated  by 
the  calculations  below,  which  correspond  to  u  light  cruiser, 
whoso  metacentric  diagram  i.s  given  In  fig.  121. 


